††thanks:  The author was supported in part by National Research Foundation of
South Korea (2022R1I1A1A01055306).

# Hilbert scheme of linearly normal curves in $\mathbb{P}^{r}$ with index of
speciality five and beyond

Changho Keem  Department of Mathematics, Seoul National University
Seoul 151-742, South Korea<EMAIL_ADDRESS>
###### Abstract.

We study the Hilbert scheme of smooth, irreducible, non-degenerate and
linearly normal curves of degree $d$ and genus $g$ in $\mathbb{P}^{r}$ ($r\geq
3$) whose complete and very ample hyperplane linear series $\mathcal{D}$ have
relatively small index of speciality $i(\mathcal{D})=g-d+r$. In particular we
determine the existence as well as the non-existence of Hilbert schemes of
linearly normal curves $\mathcal{H}^{\mathcal{L}}_{d,g,r}$ for every possible
triples $(d,g,r)$ with $i(\mathcal{D})=5$ and $r\geq 3$. We also determine the
irreducibility of the Hilbert scheme $\mathcal{H}^{\mathcal{L}}_{g+r-5,g,r}$
when the genus $g$ is near to the minimal possible value with respect to the
dimension of the projective space $\mathbb{P}^{r}$ for which
$\mathcal{H}^{\mathcal{L}}_{g+r-5,g,r}\neq\emptyset$, say $r+9\leq g\leq
r+11$. In the course of proofs of key results, we show the existence of
linearly normal curves of degree $d\geq g+1$ with arbitrarily given index of
speciality with some mild restriction on the genus $g$.

###### Key words and phrases:

Hilbert scheme, algebraic curves, linearly normal, special linear series,
index of speciality

###### 1991 Mathematics Subject Classification:

Primary 14C05, Secondary 14H10

##  1\.  An overview, motivation and preliminary set up

Let $\mathcal{H}_{d,g,r}$ be the Hilbert scheme of smooth, irreducible and
non-degenerate curves of degree $d$ and genus $g$ in $\mathbb{P}^{r}$. We
denote by $\mathcal{H}^{\mathcal{L}}_{d,g,r}$ the union of components of
$\mathcal{H}_{d,g,r}$ whose general element corresponds to a curve
$C\subset\mathbb{P}^{r}$ which is linearly normal. By abuse of terminology, we
say that a component of the Hilbert scheme $\mathcal{H}_{d,g,r}$ has index of
speciality $\alpha$ if the hyperplane series $\mathcal{D}=g^{r}_{d}$ of a
general element of the component has index of speciality $\alpha$, i.e.

$h^{1}(C,\mathcal{D})=g-d+\dim|\mathcal{D}|=\alpha\geq g-d+r.$

It is possible that there may exist a component of the Hilbert scheme
$\mathcal{H}_{d,g,r}$ which has index of speciality strictly greater than
$g-d+r$; cf. [21]. In other words, there may exist a (component of) Hilbert
scheme entirely consisting of non-linearly normal curves. However the index of
speciality of any component of $\mathcal{H}^{\mathcal{L}}_{d,g,r}$ is $g-d+r$,
which we define as the index of speciality of a Hilbert scheme of linearly
normal curves.

We recall the following modified assertion of Severi ([33]), which has been
given attention by some authors recently; cf. [5, 25].

######  Modified Assertion of Severi.

A nonempty $\mathcal{H}^{\mathcal{L}}_{d,g,r}$ is irreducible for any triple
$(d,g,r)$ in the Brill-Noether range

$\rho(d,g,r)=g-(r+1)(g-d+r)\geq 0.$

We note that the modified assertion of Severi makes sense only if the index of
speciality of Hilbert scheme of linearly normal curves
$\mathcal{H}^{\mathcal{L}}_{d,g,r}$ is non-negative. In other words,
$\mathcal{H}^{\mathcal{L}}_{d,g,r}\neq\emptyset$ only when $g-d+r\geq 0$.
Through a preliminary attempt to settle down the Modified Assertion of Severi
[33, 25], one has a quite extensive knowledge about certain basics of the
Hilbert scheme of linearly normal curves of small index of speciality
$\alpha\leq 4$, which can be summarized as follows; cf. [22, 5, 25, 23].

######  Remark 1.1.

We assume $r\geq 3$.

  * (i)

$\alpha=0$: $\mathcal{H}^{\mathcal{L}}_{g+r,g,r}\neq\emptyset$ and is
irreducible.

  * (ii)

$\alpha=1$: $\mathcal{H}^{\mathcal{L}}_{g+r-1,g,r}\neq\emptyset$ and is
irreducible for $g\geq r+1$ and is empty for $g\leq r$.

  * (iii)

$\alpha=2$: $\mathcal{H}^{\mathcal{L}}_{g+r-2,g,r}\neq\emptyset$ and is
irreducible for every $g\geq r+3$ and is empty if $g\leq r+2$.

  * (iv)

$\alpha=3$:

    1. (a)

$\mathcal{H}^{\mathcal{L}}_{g+r-3,g,r}=\emptyset$ for $g\leq r+4$ and $r\geq
5$.

    2. (b)

For $g\geq r+5$ and $r\geq 5$,
$\mathcal{H}^{\mathcal{L}}_{g+r-3,g,r}\neq\emptyset$ unless $g=r+6$ and $r\geq
10$.

    3. (c)

$\mathcal{H}^{\mathcal{L}}_{g+r-3,g,r}$ is irreducible for every $g\geq 2r+3$
and is reducible for almost all $g$ in the range $r+5\leq g\leq 2r+2$.

    4. (d)

Somewhat stronger results hold for curves in $\mathbb{P}^{3}$ and
$\mathbb{P}^{4}$. We refer [22, Remark 3.1] for details.

  * (v)

$\alpha=4$:

    1. (a)

$\mathcal{H}^{\mathcal{L}}_{g+r-4,g,r}=\emptyset$ for $g\leq r+6$, $r\geq 5$.

    2. (b)

$\mathcal{H}^{\mathcal{L}}_{g+r-4,g,r}\neq\emptyset$ for $g=r+7$ or for any
$g\geq r+10$, $r\geq 5$.

    3. (c)

$\mathcal{H}^{\mathcal{L}}_{g+r-4,g,r}\neq\emptyset$ if and only if $~{}5\leq
r\leq 8$ for $g=r+8$, $r\geq 5$ .

    4. (d)

$\mathcal{H}^{\mathcal{L}}_{g+r-4,g,r}\neq\emptyset$ if and only if $~{}5\leq
r\leq 11$ for $g=r+9$, $r\geq 5$.

    5. (e)

Furthermore, the irreducibility has been fully determined in all the peculiar
cases in (c) as well as several cases in (d).

    6. (f)

For curves in $\mathbb{P}^{3}$ and $\mathbb{P}^{4}$ with $\alpha=4$ – which
are not explicitly mentioned in the above – we refer [23, Remark 2.1] and
references therein for details.

Results mentioned in Remark 1.1 indicate that the Modified Assertion of Severi
may continue to hold even beyond the Brill-Noether range $\rho(d,g,r)\geq 0$
espeically when the index of speciality is rather small. For example, in the
case $\alpha=3$, the Brill-Noether range $\rho(g+r-3,g,r)\geq 0$ is $g\geq
3r+3$. However the irreducibility holds even for every $g\geq 2r+3$. There are
a couple of advantages in studying the Hilbert scheme of linearly normal
curves according to a given fixed index of speciality $\alpha$. One notes that
the residual series of the very ample and complete hyperplane series
$g^{r}_{d}$ corresponding to a general element of a component of
$\mathcal{H}^{\mathcal{L}}_{d,g,r}$ of index of speciality $\alpha$ has the
fixed dimension $\alpha-1$, regardless of the values of the genus $g$ and the
degree $d$. Therefore one may work more effectively in exploring out several
important properties of Hilbert schemes under consideration by studying the
family of curves in a fixed projective space $\mathbb{P}^{\alpha-1}$ induced
by the residual series of the (complete) hyperplane series.

To be more specific, in the cases $\alpha=2,3,4$, the residual series of
hyperplane series of a general element of a component of
$\mathcal{H}^{\mathcal{L}}_{d,g,r}$ are pencils, nets or webs. Therefore one
may expect to squeeze out several basic properties of the Hilbert scheme
$\mathcal{H}^{\mathcal{L}}_{d,g,r}$ such as existence, irreducibility,
gonality of a general element in each component, as well as the number of
moduli by considering the corresponding property of Hurwitz space, Severi
variety of plane curves or the Severi variety of curves on a surface of low
degree in $\mathbb{P}^{3}$. There is another advantage in studying the Hilbert
scheme $\mathcal{H}^{\mathcal{L}}_{d,g,r}$ according to its index of
speciality. Unless the genus $g$ is fairly high with respect to the degree $d$
of the projective curve in $\mathbb{P}^{r}$, a general curve in a component of
$\mathcal{H}^{\mathcal{L}}_{d,g,r}$ defined by a very ample complete
hyperplane series $\mathcal{D}$ is by no means extremal or nearly extremal
i.e. the genus $g$ is rather far below the maximal possible genus of curves of
degree $d$ in $\mathbb{P}^{r}$. However the residual series
$\mathcal{D}^{\vee}$ of $\mathcal{D}$ with respect to the canonical series may
induce an extremal or a nearly extremal curve in $\mathbb{P}^{\alpha-1}$. If
this occurs, studying (extrinsic) curves defined by the residual series
$\mathcal{D}^{\vee}$ may become relatively easier than directly handling the
original curve in $\mathbb{P}^{r}$. We will encounter this sort of phenomenon
several times in this article.

In this paper we study the Hilbert scheme of linearly normal curves with index
of speciality $\alpha=5$. Equivalently we study Hilbert schemes of non-
degenerate, smooth and linearly normal curves in $\mathbb{P}^{r}$ ($r\geq 3$)
of degree $d=g+r-5$. We determine the existence (i.e. non-emptyness) as well
as the non-existence (if any) of $\mathcal{H}^{\mathcal{L}}_{d,g,r}$ for every
triple $(d,g,r)$ with $g-d+r=5$. In order to show the non-emptiness of
$\mathcal{H}^{\mathcal{L}}_{d,g,r}$ for a given triple $(d,g,r)$, it is
sufficient to demonstrate the existence of a complete and very ample linear
series $g^{r}_{d}$ on a particular curve. We will carry this out on a suitably
chosen general $k$-gonal curve of genus $g$.

We also determine the irreducibility of $\mathcal{H}^{\mathcal{L}}_{d,g,r}$
when the genus $g$ is near to the minimal possible value $r+9$ with respect to
$r$ for which $\mathcal{H}^{\mathcal{L}}_{d,g,r}\neq\emptyset$; cf. Theorem
3.2. Happily, when $g$ is equal or near to $r+9$, the curve
$C\subset\mathbb{P}^{r}$ corresponding to a general element of a component of
$\mathcal{H}^{\mathcal{L}}_{d,g,r}$ or its residual curve $C^{\vee}$ \- which
is by definition the curve induced by the residual series of the very ample
and complete hyperplane series of $C$ \- necessarily lies on a surface of
relatively small degree. This is a consequence of a classical result which
says that an integral projective curve in $\mathbb{P}^{r}$ whose (arithmetic)
genus $g$ is in between the first and second Castelnuovo genus bound - which
we call a nearly extremal curve - necessarily lies on a surface of minimal or
near to minimal degree. By analyzing the family of such curves on these
special surfaces, one may determine the irreducibility of
$\mathcal{H}^{\mathcal{L}}_{d,g,r}$ for some particular triples $(d,g,r)$
especially when the genus $g$ is near to the minimal possible value with
respect to the dimension of the ambient projective space $\mathbb{P}^{r}$. In
carrying out this process, non-trivial results such as the the irreducibility
of the Severi variety of nodal curves on Hirzebruch surfaces plays an
important role. We will be more specific with our road map for the
determination of the irreducibility of $\mathcal{H}^{\mathcal{L}}_{d,g,r}$ in
corresponding sections.

For notation and conventions, we follow those in [3] and [4]; e.g. $\pi(d,r)$
is the maximal possible arithmetic genus of an irreducible, non-degenerate and
reduced curve of degree $d$ in $\mathbb{P}^{r}$ which is usually referred as
the first Castelnuovo genus bound. We shall refer to curves
$C\subset\mathbb{P}^{r}$ of degree $d$ whose (arithmetic) genus equals
$\pi(d,r)$ as extremal curves. $\pi_{1}(d,r)$ is the so-called the second
Castelnuvo genus bound which is the maximal possible arithmetic genus of an
irreducible, non-degenerate and reduced curve of degree $d$ in
$\mathbb{P}^{r}$ not lying on a surface of minimal degree $r-1$; cf. [16, page
99], [3, page 123]. We shall call curves $C\subset\mathbb{P}^{r}$ of degree
$d$ and (arithmetic) genus $g$ such that $\pi_{1}(d,r)<g\leq\pi(d,r)$ nearly
extremal curves. Following classical terminology, a linear series of degree
$d$ and dimension $r$ on a smooth curve $C$ is denoted by $g^{r}_{d}$. A base-
point-free linear series $g^{r}_{d}$ ($r\geq 2$) on a smooth curve $C$ is
called birationally very ample when the morphism $C\rightarrow\mathbb{P}^{r}$
induced by the $g^{r}_{d}$ is generically one-to-one onto (or is birational
to) its image curve. A base-point-free linear series $g^{r}_{d}$ on $C$ is
said to be compounded of an involution (compounded for short) if the morphism
induced by the linear series gives rise to a non-trivial covering map
$C\rightarrow C^{\prime}$ of degree $k\geq 2$. Throughout we work exclusively
over the field of complex numbers.

The organization of this paper is as follows. In the remainder of this
section, we briefly recall some other terminologies and our preliminary set
up. In the next section we list up several auxiliary results and prove a
couple of lemmas which are necessary for our study. In the third section, we
determine all the triples $(d,g,r)$ with $r\geq 3$ for which
$\mathcal{H}^{\mathcal{L}}_{g+r-5,g,r}\neq\emptyset$.

Subsequent three sections are devoted to determining the irreducibility of a
certain non-empty $\mathcal{H}^{\mathcal{L}}_{g+r-5,g,r}$. Specifically we
determine the irreducibility of $\mathcal{H}^{\mathcal{L}}_{2r+4,r+9,r}$ for
every $r\geq 3$. We stress that $\mathcal{H}^{\mathcal{L}}_{2r+4,r+9,r}$ is
the first non-empty Hilbert scheme of index of speciality $\alpha=5$ with the
smallest possible genus $g=r+9$ with respect to the dimension of the ambient
projective space $\mathbb{P}^{r}$ unless $r=4$. We also determine the
irreducibility of $\mathcal{H}^{\mathcal{L}}_{g+r-5,g,r}$ for $g=r+10$ ($r\geq
3$ and $r\neq 4$) as well as for $g=r+11$ ($r\geq 7$) and $g=r+12$ ($r\geq
10$).

In the final section we state and discuss generalized statements, which
partially covers those results for low index of specialities. The results we
state there have a certain numerical restriction. However, it extends the
existence of linearly normal curves to an arbitrarily given index of
speciality $\alpha$ such that $\alpha\geq r+1$. We also list up certain of
irreducibility results in several low genus cases which may be obtained by
utilizing similar kind techniques we use for lower index of speciality
$\alpha\leq 5$.

In the appendix, we provide proofs of several results which seem to have been
known to people rather widely; however the author could not trace explicit
sources in the literature and therefore provide proofs for the convenience of
readers.

We now briefly recall several terminologies, fundamental results and basic
frameworks for our study which are well-known; cf. [4] or [2, §1 and §2].

Let $\mathcal{M}_{g}$ be the moduli space of smooth curves of genus $g$. Given
an isomorphism class $[C]\in\mathcal{M}_{g}$ corresponding to a smooth
irreducible curve $C$, there exist a neighborhood $U\subset\mathcal{M}_{g}$ of
the class $[C]$ and a smooth connected variety $\mathcal{M}$ which is a finite
ramified covering $h:\mathcal{M}\xrightarrow{\ \ }U$, as well as varieties
$\mathcal{C}$, $\mathcal{W}^{r}_{d}$ and $\mathcal{G}^{r}_{d}$ proper over
$\mathcal{M}$ with the following properties:

  1. (1)

$\xi:\mathcal{C}\xrightarrow{\ \ }\mathcal{M}$ is a universal curve, i.e. for
every $p\in\mathcal{M}$, $\xi^{-1}(p)$ is a smooth curve of genus $g$ whose
isomorphism class is $h(p)$,

  2. (2)

$\mathcal{W}^{r}_{d}$ parametrizes the pairs $(p,L)$ where $L$ is a line
bundle of degree $d$ and $h^{0}(L)\geq r+1$ on $\xi^{-1}(p)$,

  3. (3)

$\mathcal{G}^{r}_{d}$ parametrizes the couples $(p,\mathcal{D})$, where
$\mathcal{D}$ is possibly an incomplete linear series of degree $d$ and
dimension $r$ on $\xi^{-1}(p)$.

For a complete linear series $\mathcal{E}$ on a smooth curve $C$, the residual
series $|K_{C}-\mathcal{E}|$ is denoted by $\mathcal{E}^{\vee}$. Given an
irreducible family $\mathcal{F}\subset\mathcal{G}^{r}_{d}$ with some geometric
meaning, whose general member is complete, the closure of the family
$\\{\mathcal{E}^{\vee}|\mathcal{E}\in\mathcal{F},~{}\mathcal{E}\textrm{ is
complete}\\}\subset\mathcal{W}^{g-d+r-1}_{2g-2-d}$ is denoted by
$\mathcal{F}^{\vee}$.

Let $\widetilde{\mathcal{G}}$ ($\widetilde{\mathcal{G}}_{\mathcal{L}}$,
respectively) be the union of components of $\mathcal{G}^{r}_{d}$ whose
general element $(p,\mathcal{D})$ corresponds to a very ample (very ample and
complete, respectively) linear series $\mathcal{D}$ on the curve
$C=\xi^{-1}(p)$. By recalling that an open subset of $\mathcal{H}_{d,g,r}$
consisting of elements corresponding to smooth irreducible and non-degenerate
curves is a $\mathbb{P}\textrm{GL}(r+1)$-bundle over an open subset of
$\widetilde{\mathcal{G}}$, the irreducibility of $\widetilde{\mathcal{G}}$
guarantees the irreducibility of $\mathcal{H}_{d,g,r}$. Likewise, the
irreducibility of $\widetilde{\mathcal{G}}_{\mathcal{L}}$ ensures the
irreducibility of $\mathcal{H}_{d,g,r}^{\mathcal{L}}$.

We recall the following fundamental fact regarding the scheme
$\mathcal{G}^{r}_{d}$ which is also well-known; cf. [16, 2.a] and [4, Ch. 21,
§3, 5, 6, 11, 12].

######  Proposition 1.2.

For non-negative integers $d$, $g$ and $r$, let

$\rho(d,g,r):=g-(r+1)(g-d+r)$

be the Brill-Noether number. The dimension of any component of
$\mathcal{G}^{r}_{d}$ is at least

$\lambda(d,g,r):=3g-3+\rho(d,g,r),$

hence the dimension of any component of $\mathcal{H}_{d,g,r}$ is at least

$\mathcal{X}(d,g,r):=\lambda(d,g,r)+\dim\mathbb{P}GL(r+1).$

Moreover, if $\rho(d,g,r)\geq 0$, there exists a unique component
$\mathcal{G}_{0}$ of $\widetilde{\mathcal{G}}$ which dominates
$\mathcal{M}$(or $\mathcal{M}_{g}$).

##  2\. Glossary of relevant generalities and auxiliary results

In this section we prepare a couple of lemmas and collect several facts which
are relevant to our study. Even though almost all the results we quote in this
section are quite well known, we list them up for the convenience of readers.
We first recall the following generalities regarding the Severi variety of
(nodal) curves on a Hirzebruch surface $\mathbb{F}_{e}$, i.e. a geometrically
ruled surface over $\mathbb{P}^{1}$ with invariant $e\geq 0$.

######  Definition 2.1.

  1. (i)

Given a Hirzebruch surface
$\mathbb{F}_{e}=\mathbb{P}(\mathcal{E})\stackrel{{\scriptstyle\pi}}{{\rightarrow}}\mathbb{P}^{1}$
where
$\mathcal{E}=\mathcal{O}_{\mathbb{P}^{1}}\oplus\mathcal{O}_{\mathbb{P}^{1}}(-e)$,
let $C_{0}$ be the section with $C_{0}^{2}=-e\leq 0$ and $f$ be a fibre of
$\pi$. Given a very ample linear system $\mathcal{L}=|aC_{0}+bf|$ on
$\mathbb{F}_{e}$, let

$p_{a}(\mathcal{L})=(a-1)(b-1-\frac{1}{2}ae)$

be the arithmetic genus of an integral curve belonging to $\mathcal{L}$.

  2. (ii)

Given an integer $0\leq g\leq p_{a}(\mathcal{L})$, we set
$\delta=p_{a}(\mathcal{L})-g$. We denote by

$\Sigma_{\mathcal{L},\delta}\subset\mathcal{L}=\mathbb{P}(H^{0}(X,\mathcal{L}))$

the (equi-singular) Severi variety which is the closure of the locus of
integral curves in the linear system $\mathcal{L}$ whose singular locus
consists of exactly $\delta$ nodes.

  3. (iii)

Let $\Sigma_{\mathcal{L},g}$ be the (equi-generic) Severi variety which is the
closure of the locus of integral curves of geometric genus $g$ in the linear
system $\mathcal{L}$.

We shall make use the following well-known results adopted for our specific
situation. Almost all the results quoted below are known to be true in a more
general context. Readers are advised to refer [11], which provides an
excellent treatment on Severi varieties on rational surfaces in general.

######  Remark 2.2.

  1. (i)

A general member of every irreducible component of the equi-generic Severi
variety $\Sigma_{\mathcal{L},g}$ is a nodal curve; cf. [17, Proposition 2.1],
[11, Theorem B2, p.177] and [18, pp. 105-117].

  2. (ii)

The equi-singular Severi variety $\Sigma_{\mathcal{L},\delta}$ is irreducible
of the expected dimension

$\dim|\mathcal{L}|-\delta=\frac{(a+1)(2b+2-ae)}{2}-1-\delta$

if non-empty; cf. [34, Proposition 2.11, Theorem 3.1].

Mainly for dimension count of certain families of curves or linear series
under our investigation, we make a note of the following facts regarding a
surface $S\subset\mathbb{P}^{n+1}$ of minimal degree $n\geq 2$.

######  Remark 2.3.

For an irreducible and non-degenerate surface $S\subset\mathbb{P}^{n+1}$ of
minimal degree $n$, one of the following holds; cf. [7, IV, Exercises pp
53-54].

  * (i)

$S$ is a smooth rational normal scroll, in which case we have;

$\dim|aH+bL|=\frac{1}{2}a(a+1)n+(a+1)(b+1)-1$ (1)

where $H$ (resp. $L$) is the class of a hyperplane section (resp. the class a
line of the ruling). For any reduced and irreducible curve
$C\subset\mathbb{P}^{n+1}$ of degree $d$ contained in the linear system
$|aH+bL|$, we have

$d=(aH+bL)\cdot H=na+b$ (2)

and by adjunction formula

$p_{a}(C)=\frac{1}{2}\,a\left(a-1\right)\cdot
n+\left(a-1\right)\left(b-1\right).$ (3)

Throughout, whenever we deal with a rational normal surface scroll
$S\subset\mathbb{P}^{n+1}$ of minimal degree, $H$ (resp. $L$) always denotes
the class of a hyperplane section (resp. the class a line of the ruling).

  * (ii)

$S$ is a Veronese surface in $\mathbb{P}^{5}$ in which case any reduced
irreducible curve in $S$ has even degree $2\cdot e$ having the arithmetic
genus $\frac{(e-1)(e-2)}{2}$.

  * (iii)

$S$ is a cone over a rational normal curve in a hyperplane
$\mathbb{P}^{n}\subset\mathbb{P}^{n+1}$, which is the image of the Hirzebruch
surface
$\mathbb{F}_{n}=\mathbb{P}(\mathcal{O}_{\mathbb{P}^{1}}\oplus\mathcal{O}_{\mathbb{P}^{1}}(-n))$.
We denote by $h$ (resp. $f$ ) the class in $\textrm{Pic}(\mathbb{F}_{n})$ of
the tautological bundle $\mathcal{O}_{\mathbb{F}_{n}}(1)$ (resp. of a fibre);
$f^{2}=0,f\cdot h=1,h^{2}=n,h=C_{0}+nf$. The morphism
$\mathbb{F}_{n}\rightarrow S\subset\mathbb{P}^{n+1}$ given by the complete
linear system $|h|$ is an embedding outside $C_{0}$ \- the curve with negative
self-intersection - and contracts $C_{0}$ to the vertex $P$ of the cone $S$;

$C_{0}^{2}=-n,C_{0}\in|h-nf|,K_{\mathbb{F}_{n}}=-2h+(n-2)f=-2C_{0}+(-n-2)f.$

Let $C\subset S$ be a reduced and non-degenerate curve of degree $d$ and let
$\widetilde{C}$ be the strict transformation of $C$ under the
desingularization $\widetilde{S}\cong\mathbb{F}_{n}\rightarrow S$. Setting
$k=\widetilde{C}\cdot f$, we have $\widetilde{C}\equiv kh+(d-nk)f$. The
adjunction formula gives

$p_{a}(\widetilde{C})=1/2\,\left(k-1\right)\left(2\,d-nk-2\right).$ (4)

We further remark that

$0\leq\widetilde{C}\cdot C_{0}=\widetilde{C}\cdot(h-nf)=d-nk=m$ (5)

where $m$ is the multiplicity of $C$ at the vertex $P$ of the cone $S$.

Regarding families of curves with high genus – especially nearly extremal
curves – we quote the following result which we shall use occasionally for
determining the irreducibility of a Hilbert scheme; cf. [16, Corollary 3.16,
page 100].

######  Proposition 2.4 (Harris).

Suppose $r\neq 3$ or $5$, and $\pi(d,r)\geq g>\pi_{1}(d,r)$. Then
$\mathcal{H}_{d,g,r}\neq\emptyset$ if and only if there exist integers $a>0$
and $b$ satisfying (2) and (3). Moreover, $\mathcal{H}_{d,g,r}$ has exactly
one irreducible component for each such pair $(a,b)$.

The following fact should be known to people as a folklore, which the author
does not know of any source of a proof in the literature. We provide a proof
in Appendix A.

######  Proposition 2.5.

Smooth curves in $\mathbb{P}^{n+1}$ lying on a cone over a rational normal
curve in a hyperplane $H\cong\mathbb{P}^{n}$ is a specialization of curves
lying on a rational normal surface scroll.

######  Remark 2.6.

By Proposition 2.5, a family of curves in $\mathbb{P}^{r}$ lying on cones over
rational normal curves in $H\cong\mathbb{P}^{r-1}$ is in the boundary of the
family of curves residing inside smooth surfaces of minimal degree.
Consequently, we may assume that a surface $S$ of minimal degree containing a
smooth reduced curve in $\mathbb{P}^{r}$ is smooth.

The following lemma - regarding multiples of the unique pencil $g^{1}_{k}$ on
a general $k$-gonal curve - will be used to show the existence of a complete
very ample linear series with a given index of speciality $\alpha$.

######  Lemma 2.7.

[10, Proposition 1.1] Assume $2k-g-2<0$. Let $C$ be a general $k$-gonal curve
of genus $g$, $k\geq 2$, $0\leq m$, $n\in\mathbb{Z}$ such that

$g\geq 2m+n(k-1)$ (6)

and let $D\in C_{m}$. Assume that there is no $E\in g^{1}_{k}$ with $E\leq D.$
Then $\dim|ng^{1}_{k}+D|=n$.

The following inequality - known as Castelnuovo-Severi inequality - shall be
used occasionally; cf. [1, Theorem 3.5].

######  Remark 2.8 (Castelnuovo-Severi inequality).

Let $C$ be a curve of genus $g$ which admits coverings onto curves $E_{h}$ and
$E_{q}$ with respective genera $h$ and $q$ of degrees $m$ and $n$ such that
these two coverings admit no common non-trivial factorization; if $m$ and $n$
are primes this will always be the case. Then

$g\leq mh+nq+(m-1)(n-1).$

We fix the following standard notation and convention which we shall use for
the existence of linearly normal smooth curves with prescribed degree and
genus $(d,g)$ inside a projective space of relatively low dimensions.

######  Remark 2.9.

(a) Let $\mathbb{P}^{2}_{s}$ be the rational surface $\mathbb{P}^{2}$ blown up
at $s$ general points. Let $e_{i}$ be the class of the exceptional divisor
$E_{i}$ and $l$ be the class of a line $L$ in $\mathbb{P}^{2}$. For integers
$b_{1}\geq b_{2}\geq\cdots\geq b_{s}$, let
$(a;b_{1},\cdots,b_{i},\cdots,b_{s})$ denote class of the linear system
$|aL-\sum b_{i}E_{i}|$ on $\mathbb{P}^{2}_{s}$. By abuse of notation we use
the same expression $(a;b_{1},\cdots,b_{i},\cdots,b_{s})$ for the linear
system $|aL-\sum b_{i}E_{i}|$ itself or the line bundle
$\cal{O}_{\mathbb{P}^{2}_{s}}(aL-\sum b_{i}E_{i})$. We use the notation

$(a;b_{1}^{s_{1}},\cdots,b_{j}^{s_{j}},\cdots,b_{t}^{s_{t}}),~{}\sum s_{j}=s$

when $b_{j}$ appears $s_{j}$ times consecutively in the linear system
$|aL-\sum b_{i}E_{i}|$. (b) We also use the following convention; if the
exponent $s_{j}$ of $b_{j}$ is zero in the above expression, then such an
entry does not appear in the actual linear system, i.e.

$(a;b_{1}^{s_{1}},\cdots,b_{j}^{s_{j}=0},\cdots,b_{t}^{s_{t}})=(a;b_{1}^{s_{1}},\cdots,\widehat{b_{j}^{s_{j}}},\cdots,b_{t}^{s_{t}}).$

(c) Assume $3\leq s\leq 6$. Let $C$ be a divisor on $\mathbb{P}^{2}_{s}$ and
let $(a;b_{1},\cdots,b_{s})$ be the class corresponding to
$|\mathcal{O}_{S}(C)|$. Suppose that $(a;b_{1},\cdots,b_{s})$ satisfies the
conditions

(1) $a\geq b_{1}+b_{2}+b_{3}$

(2) $a\geq\textrm{max}(0,b_{1})$

(3) $b_{1}\geq b_{2}\geq\cdots\geq b_{s}$.

Then the followings hold; cf. [27, Remark 3.1.2, Proposition 3.1.3],[19, sec.
4 proof of 4.1 and Ex 4.8] and [15, Section 2].

  1. (i)

If $C$ is reduced then $b_{s}\geq-1$.

  2. (ii)

Assume $(a;b_{1},\cdots,b_{s})\neq(a;a,0,\cdots,0)$, $a\geq 2$. A general
member of the complete linear system $|\mathcal{O}_{S}(C)|$ is smooth and
connected if and only if $b_{s}\geq 0$.

  3. (iii)

Identifying $\mathbb{P}^{2}_{s}$ with the image of the embedding
$\mathbb{P}^{2}_{s}\lhook\joinrel\xrightarrow{|-K_{\mathbb{P}^{2}_{s}}|}\mathbb{P}^{9-s}$,
a general member in $|\mathcal{O}_{S}(C)|$ is linearly normal in
$\mathbb{P}^{9-s}$ if and only if $b_{s}\geq 1$, provided

$(a;b_{1},\cdots,b_{s})\neq(\lambda+3t,\lambda+t,t,\cdots,t)\textrm{ for some
}\lambda\geq 2$

in which case $h^{1}(\mathbb{P}^{9-s},\mathcal{I}_{C}(1))=0$ if and only if
$t\geq 2$.

  4. (iv)

In Appendix B, we will give a slightly more general criteria covering (iii)
for the linear normality of a curve lying on a blown up surface of
$\mathbb{P}^{2}$, suitably embedded into $\mathbb{P}^{r}$ by a certain very
ample line bundle.

(d) For the very ampleness of a linear system $|aL-\sum b_{i}E_{i}|$, we
frequently use the main result in [13], sometimes without explicit mention.

We also make a note of the following lemma which will be used for the non-
existence (emptyness) of peculiar Hilbert schemes
$\mathcal{H}^{\mathcal{L}}_{g+r-5,g,r}$ for $g=r+10,~{}r\geq 9$ and
$g=r+11,~{}r\geq 12$.

######  Lemma 2.10.

Let $\mathcal{E}=g^{\alpha-1}_{e}$ ($\alpha\geq 4$) be a special complete
linear series on a smooth curve $C$ of genus $g\geq 5$, with (possibly empty)
base locus $\Delta$. Let
$\mathcal{E}^{\prime}:=\mathcal{E}-\Delta=g^{\alpha-1}_{e^{\prime}}$ be the
moving part of $\mathcal{E}$. We assume that either

  * (i)

$\mathcal{E}^{\prime}$ induces a double covering
$C\stackrel{{\scriptstyle\eta}}{{\rightarrow}}E$ onto a possibly singular
curve $E\subset\mathbb{P}^{\alpha-1}$ of (geometric) genus $h\geq 1$. Let
$\widetilde{E}\stackrel{{\scriptstyle\varepsilon}}{{\rightarrow}}E\subset\mathbb{P}^{\alpha-1}$
be the normalization of the curve $E$. Let
$C\stackrel{{\scriptstyle\widetilde{\eta}}}{{\rightarrow}}\widetilde{E}$ be
the morphism associated with $\eta$, i.e.
$\eta=\varepsilon\circ\widetilde{\eta}$; set

$\varepsilon^{*}(\mathcal{O}_{\mathbb{P}^{\alpha-1}}(1))=g^{\alpha-1}_{e^{\prime}/2},~{}~{}|\eta^{*}(\mathcal{O}_{\mathbb{P}^{\alpha-1}}(1))|=|\widetilde{\eta}^{*}\varepsilon^{*}(\mathcal{O}_{\mathbb{P}^{\alpha-1}}(1))|=\mathcal{E^{\prime}}.$

Suppose that the linear series $g^{\alpha-1}_{e^{\prime}/2}$ on
$\widetilde{E}$ which is pulled back to $\mathcal{E}^{\prime}$ via
$\widetilde{\eta}$, i.e.
$\widetilde{\eta}^{*}(g^{\alpha-1}_{e^{\prime}/2})=\mathcal{E^{\prime}}$ –
which is necessarily complete by the completeness of $\mathcal{E}^{\prime}$) –
is non special or

  * (ii)

$\Delta\neq\emptyset$ and $\mathcal{E}$ induces a triple covering onto a
rational curve.

Then $\mathcal{E}^{\vee}$ is not very ample.

###### Proof.

(i) For any $u\in\widetilde{E}$,

$\displaystyle\dim|\mathcal{E}+\widetilde{\eta}^{*}(u)|$ $\displaystyle=$
$\displaystyle\dim|\mathcal{E}^{\prime}+\Delta+\widetilde{\eta}^{*}(u)|=\dim|\widetilde{\eta}^{*}(g^{\alpha-1}_{e^{\prime}/2}+u)+\Delta|$
$\displaystyle=$
$\displaystyle\dim|\widetilde{\eta}^{*}(g^{\alpha}_{e^{\prime}/2+1})+\Delta|\geq\dim\mathcal{E}+1$

and hence $\mathcal{E}^{\vee}$ is not very ample. (ii)) Let $g^{1}_{3}$ be the
(unique) trigonal pencil on $C$. Choose $q\in\Delta$ and let $q+t+s\in
g^{1}_{3}$ be the unique trigonal divisor containing $q$. We then have

$\displaystyle\dim|\mathcal{E}+t+s|$ $\displaystyle=$
$\displaystyle\dim|(\alpha-1)g^{1}_{3}+\Delta+t+s|$ $\displaystyle=$
$\displaystyle\dim|(\alpha-1)g^{1}_{3}+(q+t+s)+(\Delta-q)|$ $\displaystyle=$
$\displaystyle\dim|\alpha g^{1}_{3}+(\Delta-q)|\geq\dim\mathcal{E}+1,$

hence $\mathcal{E}^{\vee}$ is not very ample. ∎

On the contrary, the following lemma will be used for the existence of
$\mathcal{H}^{\mathcal{L}}_{g+r-5,g,r}$ with $g=r+12$ when $r$ is big enough.
The proof is lengthy which we provide in Appendix C.

######  Lemma 2.11.

Let $C\stackrel{{\scriptstyle\eta}}{{\rightarrow}}E$ be a triple covering of
an elliptic curve $E$ of genus $g\geq 3\alpha+7$, $\alpha\geq 3$. For a
complete $g^{\alpha-1}_{\alpha}\in W^{\alpha-1}_{\alpha}(E)$, we put
$\mathcal{E}:=|\eta^{*}(g^{\alpha-1}_{\alpha})|$. Then
$\dim\mathcal{E}=\alpha-1$ and
$\mathcal{E}^{\vee}=|K_{C}-\mathcal{E}|=g^{g-2\alpha-2}_{2g-2-3\alpha}$ is
very ample.

We close this section with an immediate consequence of Lemma 2.11.

######  Corollary 2.12.

For $\alpha\geq 3$, $r\geq\alpha+5$

$\mathcal{H}^{\mathcal{L}}_{2r+\alpha+2,r+2\alpha+2,r}\neq\emptyset.$

###### Proof.

We take the linear series
$\mathcal{E}=g^{\alpha-1}_{3\alpha}=\eta^{*}(g^{\alpha-1}_{\alpha})$ on a
triple covering of an elliptic curve
$C\stackrel{{\scriptstyle\eta}}{{\rightarrow}}E$. Since $g=r+2\alpha+2\geq
3\alpha+7$, the complete linear series
$\mathcal{E}^{\vee}=g^{r}_{2r+\alpha+2}$ is very ample by Lemma 2.11. ∎

##  3\. Existence & non-existence of linearly normal curves with the index of
speciality $\alpha=5$

This section is devoted to the existence of linearly normal curves in
$\mathbb{P}^{r\geq 3}$ with index of speciality $\alpha=5$. We determine all
the triples $(g+r-5,g,r)$ such that
$\mathcal{H}^{\mathcal{L}}_{g+r-5,g,r}\neq\emptyset$.

Before proving the first main result of this section, we make a remark
concerning the existence of linearly normal smooth curves which directly comes
from well-known results in Brill-Noether theory.

######  Remark 3.1.

(i) In the Brill-Noether range

$\rho(d,g,r)=\rho(d,g,r)=g-(r+1)(g-d+r)\geq 0$

with $r\geq 3$, the existence of a smooth linearly normal curve in
$\mathbb{P}^{r}$ of degree $d$ and genus $g$ follows from a theorem due to
Eisenbud-Harris [16, Theorem (1.8)]; a general element of $W^{r}_{d}(C)$ on a
general curve $C$ of genus $g$ is very ample, which is also complete by the
fact that no component of $W^{r}_{d}(C)$ is entirely contained in
$W^{r+1}_{d}(C)$ (for any $C$) [3, Lemma (3.5), Chapt. IV]. Hence we see that

$\mathcal{H}^{\mathcal{L}}_{d,g,r}\neq\emptyset\textrm{ if }\rho(d,g,r)\geq
0\textrm{ and }r\geq 3.$

(ii) Therefore we may focus on triples $(d,g,r)$ which are outside the Brill-
Noether range. In our current situation $g-d+r=5$, we are particularly
interested in the range

$\rho(d,g,r)=g-5(r+1)<0.$

The following theorem provides the full list of triples $(d,g,r)$ with
$g-d+r=5$ and $r\geq 3$ for which
$\mathcal{H}^{\mathcal{L}}_{g+r-5,g,r}\neq\emptyset$.

######  Theorem 3.2.

  1. (a)

$\mathcal{H}^{\mathcal{L}}_{g+r-5,g,r}=\mathcal{H}_{g+r-5,g,r}=\emptyset\textrm{
for }g\leq r+8,r\neq 4$.

For $r=4$,
$\mathcal{H}^{\mathcal{L}}_{g+r-5,g,r}=\mathcal{H}^{\mathcal{L}}_{g-1,g,4}=\emptyset$
if $g\leq r+7=11$ and

$\mathcal{H}^{\mathcal{L}}_{g-1,g,4}\neq\emptyset$ for $g=r+8=12$.

  2. (b)

$\mathcal{H}^{\mathcal{L}}_{g+r-5,g,r}\neq\emptyset$ for $g=r+9$, $r\geq 3$.

  3. (c)

$\mathcal{H}^{\mathcal{L}}_{g+r-5,g,r}=\emptyset$ for $g=r+10$ if and only if
$r\geq 9$.

  4. (d)

$\mathcal{H}^{\mathcal{L}}_{g+r-5,g,r}=\emptyset$ for $g=r+11$ if and only if
$r\geq 12$.

  5. (e)

$\mathcal{H}^{\mathcal{L}}_{g+r-5,g,r}\neq\emptyset$ for $g\geq r+12$, $r\geq
3$.

###### Proof.

  1. (a-i)

$r=3$: If $g\leq 10$ one has $\pi(g-2,3)\lneq g$ which is an absurdity. If
$g=11$ we have $\pi_{1}(g-2,3)=10<g\leq\pi(9,3)=12$ and hence $C$ is a nearly
extremal curve lying on a quadric surface in $\mathbb{P}^{3}$. However there
is no pair $(a,b)\in\mathbb{N}\times\mathbb{N}$ satisfying $g=11=(a-1)(b-1)$
and $d=g-2=9=a+b$.

  2. (a-ii)

$r=4$: If $g\leq r+7=11$, $g\leq\pi(g-1,4)\leq g-2$ a contradiction. A non-
degenerate irreducible curve of degree $d=11$ in $\mathbb{P}^{4}$ has maximal
arithmetic genus $\pi(11,4)=12=r+8$ and such an extremal curve exists in the
linear system $|4H-L|$ on a cubic surface scroll in $\mathbb{P}^{4}$; cf. [3,
III, Theorem 2.5].

  3. (a-iii)

$r=5$: For $g\leq 13=r+8$, one has $\pi(g,5)\leq g-1$, a contradiction.

  4. (a-iv)

$r\geq 6$: One computes $m:=[\frac{g+r-6}{r-1}]\leq 2$ if $r\geq 6$ and $g\leq
r+8$. In case $m=2$, one has

$g\leq\pi(g+r-5,r)=\frac{m(m-1)}{2}(r-1)+m(g+r-6-2(r-1))=2g-r-9$

which is not compatible with the assumption $g\leq r+8$. Same contradiction
occurs when $m=1$.

  1. (b-i)

$r=3$: For $g=r+9=12$, there exists a smooth curve of degree $d=10$ of type
$(3,7)$ on a smooth quadric in $\mathbb{P}^{3}$. There also exists a smooth
curve in $\mathbb{P}^{3}$ with $(d,g)=(10,12)$ on a non-singular cubic
surface; a proper transformation of a plane curve of degree $9$ with $5$
ordinary triple points and a node on $\mathbb{P}^{2}$ blown up at the $6$
assigned singular points; $C\in(9;3^{5},2)$ on $\mathbb{P}^{2}_{6}$.

  2. (b-ii)

$r=4$: On a general trigonal curve of genus $g=13$,
$|K-4g^{1}_{3}|=g^{4}_{12}$ is very ample; we may take $m=2,k=3,n=4$ in Lemma
2.7. Such a curve exists on a smooth cubic surface scroll in $\mathbb{P}^{4}$
in the linear system $|3H+3L|$. There is another irreducible family of curves
lying on a del Pezzo surface $\mathbb{P}^{2}_{5}\subset\mathbb{P}^{4}$;
$C\in(9;3^{5})$. Both curves are linearly normal; if not, $C$ is an image of a
projection from a curve $\tilde{C}\subset\mathbb{P}^{5}$ of the same degree
$d=12$, contradicting the Castelnuovo’s genus bound $\pi(12,5)=10>g=r+9=13$.
In the next section we will see that these two types of curves form the only
two components of $\mathcal{H}^{\mathcal{L}}_{12,13,4}$; cf. Proposition 4.1
(v).

  3. (b-iii)

$r=5$: Similarly there exists a smooth linearly normal curve of degree
$d=g=14$ on a smooth quartic surface scroll $S\subset\mathbb{P}^{5}$. We may
take $C\in|3H+2L|$ which is trigonal. Note that by adjunction,

$g^{5}_{14}=|K_{C}-4g^{1}_{3}|=|K_{S}+C-4L|_{|C}=|H|_{|C},$

which is complete and hence $C$ is linearly normal. Alternatively since
$\pi(14,6)=11<g$, every smooth curve $C\subset\mathbb{P}^{5}$ with
$(d,g)=(14,14)$ is linearly normal.

  4. (b-iv)

$r\geq 6$: For $g=r+9$ we have $\pi(g+r-5,r)=r+9$ and the curve corresponding
to a general element of a component of $\mathcal{H}_{2r+4,r+9,r}$ is an
extremal curve. An extremal curve is always linearly normal and hence

$\mathcal{H}^{\mathcal{L}}_{2r+4,r+9,r}=\mathcal{H}_{2r+4,r+9,r}\neq\emptyset.$

(c) We first show that $\mathcal{H}^{\mathcal{L}}_{g+r-5,g,r}=\emptyset$ when
$g=r+10$ and $r\geq 9$. Let $\mathcal{D}\in\mathcal{G}$ be a complete very
ample $g^{r}_{g+r-5}$ and set $\mathcal{E}:=\mathcal{D}^{\vee}=g^{4}_{13}$.
Since

$\pi(13,4)=18<19\leq r+10=g,$

$\mathcal{E}$ is compounded with non-empty base locus $\Delta$,
$\deg\Delta=\delta\geq 1$ inducing a $k$-sheeted map $\eta:C\rightarrow E$
onto a possibly singular non-degenerate curve $E\subset\mathbb{P}^{4}$, $\deg
E=f$. The following triples $(k,f,\delta)$ are possible subject to the
conditions

$k\geq 2,~{}\deg E=f\geq\dim\mathbb{P}^{4},~{}k\cdot f+\delta=13.$

  1. (1)

$(k,f,\delta)=(2,6,1)$; $C\stackrel{{\scriptstyle\eta}}{{\rightarrow}}E$ is a
double cover of a curve $E$ of genus $2$ and
$\mathcal{E}=\eta^{*}(g^{4}_{6})+\Delta$.

  2. (2)

$(k,f,\delta)=(2,5,3)$; $C\stackrel{{\scriptstyle\eta}}{{\rightarrow}}E$ is a
double cover of a curve $E$ of genus $1$ and
$\mathcal{E}=\eta^{*}(g^{4}_{5})+\Delta$.

  3. (3)

$(k,f,\delta)=(2,4,5)$; $C$ is hyperelliptic.

  4. (4)

$(k,f,\delta)=(3,4,1)$; $C$ is trigonal and $\mathcal{E}=4g^{1}_{3}+\Delta$.

Note that linear series $\widetilde{\mathcal{E}}$ on $E$ (or on the
normalization of $E$) such that
$\eta^{*}(\widetilde{\mathcal{E}})+\Delta=\mathcal{E}$ is complete since
$\mathcal{E}$ is complete. $\widetilde{\mathcal{E}}$ is non-special by
Clifford’s theorem.

In the cases (1), (2), (4), the residual series
$\mathcal{D}=\mathcal{E}^{\vee}$ is not very ample by Lemma 2.10; we may take
$\alpha=5,e=13$ in Lemma 2.10. In the case (3), we recall that a hyperelliptic
curve does not carry a special very ample linear series. Hence
$\mathcal{H}^{\mathcal{L}}_{g+r-5,r+10,r}=\emptyset$ if $r\geq 9$.

We next show

$\mathcal{H}^{\mathcal{L}}_{2r+5.r+10.r}=\mathcal{H}_{2r+5,r+10,r}\neq\emptyset~{}\textrm{
for }3\leq r\leq 8.$

We consider $\mathbb{P}^{2}_{9-r}\subset\mathbb{P}^{r}$ which may be realized
as a smooth surface in $\mathbb{P}^{r}$ embedded by the anti canonical linear
system $|-K_{\mathbb{P}^{2}_{9-r}}|=(3,1^{9-r})$. Set

$\mathcal{L}(\beta,r):=(8;3,2^{8-r-\beta},1^{\beta});~{}~{}0\leq\beta\leq
8-r.$ (7)

Let $C\in\mathcal{L}(\beta,r)$ be a general member. Since
$\mathcal{L}(\beta,r)$ is very ample by [13, Theorem] or [19, V, 4.13],
$C\subset PP^{r}$ is smooth of degree $d$ and genus $g$ where

$\displaystyle
d=\mathcal{L}(\beta,r)\cdot(-K_{\mathbb{P}^{2}_{9-r}})=(8;3,2^{8-r-\beta},1^{\beta})\cdot(3;1^{9-r})=2r+5+\beta,$
(8) $\displaystyle g=\frac{(8-1)(8-2)}{2}-3-(8-r-\beta)=r+10+\beta.$ (9)

We take $\beta=0$, then

$\mathcal{L}(0,r)=(8;3,2^{8-r},1^{0})=(8;3,2^{8-r}),$

according to the convention we employed in Remark 2.9 (b).

Note that a smooth curve of degree $d=2r+5$ and genus $g=r+10$ in
$\mathbb{P}^{r}$ is linearly normal since

$\pi(2r+5,r+1)<g=r+10;$

i.e. if it is not linearly normal then one may embed the curve in a higher
dimensional projective space $\mathbb{P}^{r+1}$ and up there $g=r+10$ is too
big for the genus of a smooth curve of degree $d=2r+5$. Alternatively, one may
use Remark 2.9 (c)(iii).

(d) If $g=r+11$ and $r\geq 12$ – as in the first half of the previous case (c)
($g=r+10$ and $r\geq 9$) – we have

$g=r+11\geq 23>\pi(14,4)=22$

and hence $\mathcal{E}$ is compounded with (possibly empty) base locus
$\Delta$ inducing a $k^{\geq 2}$ – sheeted map onto a curve of degree $f$ in
$\mathbb{P}^{4}$. Hence Lemma 2.10 applies and we may conclude
$\mathcal{H}^{\mathcal{L}}_{g+r-5.r+11.r}=\emptyset$ for $r\geq 12$. We omit
the details.

In what follows, we show
$\mathcal{H}^{\mathcal{L}}_{g+r-5,g,r}=\mathcal{H}^{\mathcal{L}}_{2r+6,r+11,r}\neq\emptyset$
for $3\leq r\leq 11$.

  1. (d-i)

We first deal with the cases $3\leq r\leq 7$ using the same surface
$\mathbb{P}^{2}_{9-r}\subset\mathbb{P}^{r}$ embedded by
$|-K_{\mathbb{P}^{2}_{9-r}}|$. We take $\beta=1$ in the linear system (7)

$\mathcal{L}(\beta,r)=(8;3,2^{8-r-\beta},1^{\beta})$

and obtain

$\mathcal{H}_{2r+6,r+11,r}\neq\emptyset~{}\mathrm{if}~{}3\leq r\leq 7.$

Note that the obvious condition $8-r-\beta\geq 0$ on one of the exponent in
$\mathcal{L}(\beta,r)$ forces $r\leq 7$ if $\beta=1$. If $5\leq r\leq 7$, a
smooth curve of degree $d=2r+6$ and genus $g=r+11$ in $\mathbb{P}^{r}$ is
linearly normal since

$g=r+11>\pi(2r+6,r+1)$

and hence

$\mathcal{H}_{2r+6,r+11,r}=\mathcal{H}^{\mathcal{L}}_{2r+6,r+11,r}\neq\emptyset~{}\mathrm{if}~{}5\leq
r\leq 7.$

For $r=3,4$ we have

$\mathcal{H}^{\mathcal{L}}_{2r+6,r+11,r}\neq\emptyset$

by Remark 2.9 (c)(iii).

  2. (d-ii)

For the cases $8\leq r\leq 11$, we take the non-singular surface $S$ as

$\mathbb{P}^{2}_{12-r}\lhook\joinrel\xrightarrow{|(4;2,1^{11-r})|}S\subset\mathbb{P}^{r},$

embedded by the very ample111 $(4;2,1^{11-r})$ is very ample even for $4\leq
r\leq 11.$ linear system $(4;2,1^{11-r})$; cf. [13, Theorem]. We consider a
linear system

$\mathcal{M}(\gamma,r)=(9;4,2^{11-r-\gamma},1^{\gamma}),~{}~{}0\leq\gamma\leq
11-r$ (10)

and let $C\in\mathcal{M}(\gamma,r)$ be a general member. Since
$\mathcal{M}(\gamma,r)$ is very ample by [13, Theorem], $C$ is smooth of
degree $d$ and genus $g$ in $\mathbb{P}^{r}$ where

$\displaystyle d=\mathcal{M}(\gamma,r)\cdot(4;2,1^{11-r})=2r+6+\gamma,$ (11)
$\displaystyle g=\frac{(9-1)(9-2)}{2}-\frac{4\cdot
3}{2}-(11-r-\gamma)=r+11+\gamma.$ (12)

For the particular value $\gamma=0$, we have

$\mathcal{M}(0,r)=(9;4,2^{11-r}).$

Note that

$g=r+11>\pi(2r+6,r+1),$

holds222This strict inequality holds for $5\leq r\leq 11$. for $8\leq r\leq
11$ and hence a general member in $\mathcal{M}(0,r)$ is linearly normal of
degree $d=2r+6$ and genus $g=r+11$ in $\mathbb{P}^{r}$. Therefore we may claim

$\mathcal{H}^{\mathcal{L}}_{2r+6,r+11,r}\neq\emptyset\textrm{ for }8\leq r\leq
11.$

(e) In order to avoid unnecessary and complicated computation we first show

(e-i) $\mathcal{H}^{\mathcal{L}}_{g+r-5,g,r}\neq\emptyset$ for $g=r+12$,
$r\geq 3$ and then proceed to show

(e-ii) $\mathcal{H}^{\mathcal{L}}_{g+r-5,g,r}\neq\emptyset$ for $g\geq r+13$,
$r\geq 3$.

  1. (e-i-1)

$g=r+12$, $3\leq r\leq 6$: We take $\beta=2$ in the linear system (7),

$\mathcal{L}(2,r)=(8;3,2^{6-r},1^{2})$

containing a smooth curve of degree $d=2r+7$ and genus $g=r+12$ by (8), (9).
The linear normality follows from Remark 2.9 (c).

  2. (e-i-2)

$g=r+12,\,7\leq r\leq 10$: We take the linear system $\mathcal{M}(\gamma,r)$
in (10) on the surface $\mathbb{P}^{2}_{12-r}\subset\mathbb{P}^{r}$ and put
$\gamma=1$. A general member of the very ample linear system

$\mathcal{M}(1,r)=(9;4,2^{10-r},1)$

is smooth of degree $d=2r+7$ and genus $g=r+12$ in $\mathbb{P}^{r}$ by (11),
(12). Assume that a general $C\in\mathcal{M}(1,r)$ is not linearly normal. Let
$\widetilde{C}\subset\mathbb{P}^{r+1}$ be a curve of degree $d=2r+7$ which is
projected onto $C\subset\mathbb{P}^{r}$ from a point outside $\widetilde{C}$.
Note that

$g=r+12=\pi(2r+7,r+1)$

for $7\leq r\leq 10$ and hence $\widetilde{C}\subset\mathbb{P}^{r+1}$ is an
extremal curve lying on surface of minimal degree in $\mathbb{P}^{r+1}$. In
case $\widetilde{C}$ lies on a smooth scroll,

$\widetilde{C}\in|(m+1)H-(r+1-\varepsilon-2)L|=|3H-(r-7)L|$

where $m:=[\frac{d-1}{r}]=2,\varepsilon=d-1-mr$; cf. [3, Theorem 2.5, p122].
Hence $C\cong\widetilde{C}$ is trigonal and the trigonal pencil is cut out by
the rulings of the scroll. On the other hand, a general $C\in\mathcal{M}(1,r)$
has a base-point-free $g^{1}_{5}$ cut out by lines through the $4$-fold point
on the plane model of $C$. By the Castelnuovo-Severi inequality,
$g\leq(3-1)\cdot(5-1)=8$, a contradiction. If
$\widetilde{C}\subset\mathbb{P}^{r+1}$ lies on a rational normal cone, we put
$n=r$, $d=2r+7$, $p_{a}(\widetilde{C})=r+12$ in Remark 2.3 (iii) (in the genus
formula (4)) to get $k=3$ hence $\widetilde{C}$ is trigonal leading to the
same contradiction.

  3. (e-i-3)

$g=r+12,r\geq 10$; we take $\alpha=5$ in Corollary 2.12 and may conclude
$\mathcal{H}^{\mathcal{L}}_{2r+7,r+12,r}\neq\emptyset$.

  4. (e-ii)

$g\geq r+13$: We are primarily interested in the existence of linearly normal
curves in $\mathbb{P}^{r}$ with index of speciality $\alpha=5$. However we
will show the existence of linearly normal curves in $\mathbb{P}^{r}$ with any
given index of speciality $\alpha\geq 2$ under an additional condition
$r\geq\alpha+1$ together with a mild restriction on the genus $g$. We use
Lemma 2.7 regarding the dimension of the linear series which is a multiple of
$g^{1}_{k}$ on general $k$-gonal curves. Our assertion

$\mathcal{H}^{\mathcal{L}}_{g+r-5,g,r}\neq\emptyset\textrm{~{}for~{} }g\geq
r+13$

is an immediate consequence of the following claim when $r\geq 6$.

  5. (e-ii-1)

Claim (3.2.A): Fix an integer $\alpha\geq 3$. For every $r\geq\alpha+1$ and
$g\geq r+3\alpha-2$,

$\mathcal{H}^{\mathcal{L}}_{g+r-\alpha,g,r}\neq\emptyset.$

We set

$e:=g-r+\alpha-2=\deg|K_{C}-\mathcal{D}|;~{}\mathcal{D}\in\mathcal{G}\subset\widetilde{\mathcal{G}}_{\mathcal{L}}\subset\mathcal{G}^{r}_{g+r-\alpha}$

and

$e\equiv\sigma~{}(\textrm{mod}~{}\alpha-1),~{}~{}0\leq\sigma\leq\alpha-2.$

By the assumption $g\geq r+3\alpha-2$, we have

$e=g-r+\alpha-2=(\alpha-1)k+\sigma\geq 4(\alpha-1)$ (13)

for some $k\geq 4$. Since $r\geq\alpha+1$, we have

$\displaystyle\rho(k,g,1)$
$\displaystyle=2k-g-2=-(\alpha-3)k-r-\sigma+\alpha-4$
$\displaystyle\leq-(\alpha-3)k-(\alpha+1)-\sigma+\alpha-4$
$\displaystyle=-(\alpha-3)k-\sigma-5\lneqq 0$

and hence $\mathcal{M}^{1}_{g,k}\subsetneq\mathcal{M}_{g}$.

Let $C\in\mathcal{M}^{1}_{g,k}$ be a general $k$-gonal curve. We choose
$q_{1}+\cdots+q_{\sigma}\in C_{\sigma}$ such that no pair $(q_{i},q_{j})$,
$q_{i}\neq q_{j}$ are in the same fibre of the $k$-sheeted map onto
$\mathbb{P}^{1}$ defined by the unique $g^{1}_{k}$ on $C$. For an arbitrary
choice $t+s\in C_{2}$, we take

$D=q_{1}+\cdots+q_{\sigma}+t+s\in C_{\sigma+2},m=\sigma+2\textrm{ and
}n=\alpha-1$

in Lemma 2.7. By the choice of $q_{1}+\cdots+q_{\sigma}\in C_{\sigma}$, there
is no $E\in g^{1}_{k}$ such that $E\leq D$ . Furthermore the numerical
condition (6) in Lemma 2.7 is satisfied; via direct computation together with
the condition333The numerical condition (6) in Lemma 2.7 holds when
$r\geq\sigma+3$ which automatically holds if $r\geq\alpha+1$. $r\geq\alpha+1$.
Hence

$\dim|(\alpha-1)g^{1}_{k}+D|=\alpha-1$

by Lemma 2.7, which implies

$|K_{C}-(\alpha-1)g^{1}_{k}-(q_{1}+\cdots+q_{\sigma})|=g^{r}_{2g-2-k(\alpha-1)-\sigma}=g^{r}_{g+r-\alpha}$

is very ample (and complete), finishing the proof of the claim.

  6. (e-ii-2)

$3\leq r\leq 5$: If $r=3$, the existence of smooth linearly normal curves of
degree $d=g+r-5=g-2$, $g\geq r+13=16$ in $\mathbb{P}^{3}$ follows from works
of Gruson-Peskine [15] and [12]. The existence of smooth linearly normal
curves of degree $d=g+r-5=g-1$ ($d=g$ resp.) with $g\geq r+13=17$ ($g\geq 18$
resp.) in $\mathbb{P}^{4}$ ($\mathbb{P}^{5}$ resp.) follows from the work of
Rathman [32]. However, we will show the existence and the linear normality of
such curves through explicit examples we have seen in this proof.

By Remark 3.1 (ii) we only need to check the existence of linearly normal
curves outside the Brill-Noether range; $\rho(g+r-5,g,r)<0$, $3\leq r\leq 5$,
i.e.

$\begin{cases}r+13=16\leq g<(r+1)(g-d+r)=20;&r=3\\\ r+13=17\leq
g<(r+1)(g-d+r)=25;&r=4\\\ r+13=18\leq g<(r+1)(g-d+r)=30;&r=5.\end{cases}$

(i) $r=3$: The very ample linear system (7)

$\mathcal{L}(\beta,r)=(8;3,2^{8-r-\beta},1^{\beta})$

with $r=3$ and $3\leq\beta\leq 5$ on
$\mathbb{P}^{2}_{9-r}=\mathbb{P}^{2}_{6}\lhook\joinrel\xrightarrow{|(3;1^{6}|}\mathbb{P}^{3}$
covers $16\leq g=r+10+\beta\leq 18$; cf. (8), (9). For the case $g=19$, we use
the very ample linear system $(10;6,2^{2},1^{3})$ on
$\mathbb{P}^{2}_{9-r}=\mathbb{P}^{2}_{6}\lhook\joinrel\xrightarrow{|(3;1^{6}|}\mathbb{P}^{3}$.
Linear normality follows from Remark 2.9 (c).

(ii) $r=4$: Let $S\subset\mathbb{P}^{4}$ be a Bordiga surface in
$\mathbb{P}^{4}$, which is the rational surface $\mathbb{P}^{2}_{10}$ blown up
at $10$ general points and embedded into $\mathbb{P}^{4}$ by the linear system
$(4;1^{10})$. Consider

$\mathcal{B}(\delta)=(9;3,2^{\delta},1^{9-\delta})=(4;1^{10})+(5;2,1^{\delta},0^{9-\delta});1\leq\delta\leq
8,$

which is a sum of a very ample and a base-point-free linear system. Hence
$\mathcal{B}(\delta)$ is very ample whose general member is an irreducible,
non-singular curve $C$ of degree $d$ and genus $g$ where

$d=(9;3,2^{\delta},1^{9-\delta})\cdot(4;1^{10})=24-\delta,$
$g=\frac{(9-1)(9-2)}{2}-\frac{3\cdot
2}{2}-\delta=25-\delta,~{}~{}1\leq\delta\leq 8.$

A general $C\in(9;3,2^{\delta},1^{9-\delta})$ is linearly normal by Corollary
B.2. (iii) $r=5$: We may find a smooth curve of degree $d=g$ in
$\mathbb{P}^{5}$ on a Bordiga surface
$\mathbb{P}^{2}_{9}\stackrel{{\scriptstyle(4;1^{9})}}{{\hookrightarrow}}\mathbb{P}^{5}$
as follows.

     * (1)

$18\leq g=d\leq 21$: Take

$C\in(10;3^{5},2^{\delta-1},1^{5-\delta}),~{}1\leq\delta\leq 4$

on $\mathbb{P}^{2}_{9}$. Note that

$\displaystyle(10;3^{5},2^{\delta-1},1^{5-\delta})$ $\displaystyle=$
$\displaystyle(4;1^{9})+(6;2^{5},1^{\delta-1},0^{5-\delta})$ $\displaystyle=$
$\displaystyle(4;1^{9})+(3;1^{5},0^{4})+(3;1^{5},1^{\delta-1},0^{5-\delta}),$

where $(4;1^{9})$ is very ample, both $(3;1^{5},0^{4})$ and
$(3;1^{5},1^{\delta-1},0^{5-\delta})$ are base-point-free hence
$(10;3^{5},2^{\delta-1},1^{5-\delta})$ is very ample containing an irreducible
and non-singular curve $C$ as a general member by Bertini. One computes

$\displaystyle p_{a}(C)$ $\displaystyle=$
$\displaystyle\frac{1}{2}(C+K_{\mathbb{P}^{2}_{9}})\cdot C+1$ $\displaystyle=$
$\displaystyle\frac{1}{2}((10;3^{5},2^{\delta-1},1^{5-\delta})+(-3;-1^{9}))\cdot(10;3^{5},2,^{\delta-1},1^{5-\delta})+1$
$\displaystyle=$
$\displaystyle\frac{1}{2}(7;2^{5},1^{\delta-1},0^{5-\delta})\cdot(10;3^{5},2^{\delta-1},1^{5-\delta})+1=22-\delta$

and

$d=(10;3^{5},2^{\delta-1},1^{5-\delta})\cdot(4;1^{9})=22-\delta.$

$C$ is linearly normal by Appendix C .

     * (2)

$22\leq g=d\leq 25$: Take

$C\in(12;6,4,3^{3},2^{4-\delta},1^{\delta}),1\leq\delta\leq 4.$

By the same reason as the previous one,

$\displaystyle(12;6,4,3^{3},2^{4-\delta},1^{\delta})$
$\displaystyle=(4;1^{9})+(8;5,3,2^{3},1^{4-\delta},0^{\delta})$
$\displaystyle=(4;1^{9})+(5;3,2,1^{7-\delta},0^{\delta})+(3;2,1^{4},0^{4})$

is very ample; $(5;3,2,1^{7-\delta})$ ($(3;2,1^{4})$ resp.) is base-point-free
on $\mathbb{P}^{2}_{9-\delta}$ ($\mathbb{P}^{2}_{5}$ resp.) by [13]. Hence
$(5;3,2,1^{7-\delta},0^{\delta})$ ($(3;2,1^{4},0^{4})$ resp.) is base-point-
free on $\mathbb{P}^{2}_{9}$. An easy calculation yields

$d=g=21+\delta,~{}1\leq\delta\leq 4.$

     * (3)

$26\leq g=d\leq 29$: Taking

$C\in(14;8,4^{3},3,2^{4-\delta},1^{\delta}),$

we have

$\displaystyle(14;8,4^{3},3,2^{4-\delta},1^{\delta})$
$\displaystyle=(4;1^{9})+(10;7,3^{3},2,1^{4-\delta},0^{\delta})$
$\displaystyle=(4;1^{9})+(6;4,2^{3},1^{5-\delta},0^{\delta})+(4;3,1^{4},0^{4}).$

Likewise, the second and third summands are base-point-free on
$\mathbb{P}^{2}_{9}$ and hence $(14;8,4^{3},3,2^{4-\delta},1^{\delta})$ is
very ample whose general member $C$ has genus $g=25+\delta~{}(1\leq\delta\leq
4)$ and degree $d=g$ in $\mathbb{P}^{5}$.

∎

Before closing the section, we would like to make the following remark which
is worthy of being mentioned.

######  Remark 3.3.

  * (i)

The peculiar and seemingly artificial linear systems $\cal{L}(\beta,r)$,
$\cal{M}(\gamma,r)$ in (7), (10) on del Pezzo surfaces which appear in the
proof of Theorem 3.2 arise naturally by analyzing nearly extremal curves
induced by $\mathcal{E}=\mathcal{D}^{\vee}$ corresponding to a general element
$\mathcal{D}\in\mathcal{G}\subset\mathcal{G}^{r}_{g+r-5}$. More systematic
treatment appears in the course of the proof of Theorem 5.1 when we check the
irreducibility of $\mathcal{H}^{\mathcal{L}}_{2r+5,r+10,r}$.

  * (ii)

Hilbert schemes $\mathcal{H}^{\mathcal{L}}_{2r+10,r+5,r}$ and
$\mathcal{H}^{\mathcal{L}}_{2r+11,r+6,r}$ are non-empty only for low $r$. For
$\alpha=3,4$, there are also such Hilbert schemes
$\mathcal{H}^{\mathcal{L}}_{g+r-\alpha,g,r}$; cf. [22, Proposition 3.3], [23,
Theorem 2.4]. However this phenomenon is not so unusual under our setting –
i.e. studying smooth projective curves in connection with the index of
speciality – which we will see more explicitly in the final section; cf.
Peoposition 7.5.

##  4\. Irreducibility of $\mathcal{H}^{\mathcal{L}}_{g+r-5,g,r}$ for lowest
possible genus $g$

In this section, we determine the irreducibility of
$\mathcal{H}^{\mathcal{L}}_{g+r-5,g,r}$ when the genus $g$ has the lowest
possible value with respect to $r$. Recall that by Theorem 3.2 (a), (b), we
know

  * •

$\mathcal{H}^{\mathcal{L}}_{g+r-5,g,r}=\mathcal{H}_{g+r-5,g,r}=\emptyset\textrm{
for }g\leq r+8,r\neq 4$ and
$\mathcal{H}^{\mathcal{L}}_{g+r-5,g,r}\neq\emptyset$ for $g=r+9$, $r\geq 3$.

  * •

For $r=4$,
$\mathcal{H}^{\mathcal{L}}_{g+r-5,g,r}=\mathcal{H}^{\mathcal{L}}_{g-1,g,4}=\emptyset$
if $g\leq r+7=11$ and $\mathcal{H}^{\mathcal{L}}_{g-1,g,4}\neq\emptyset$ for
$g=r+8=12$.

In general, determining the irreducibility of Hilbert schemes of smooth curves
is rather a hard task. However one may obtaine the following irreducibility
result when the genus $g$ has the minimal possible value $g=r+9$ ($r\neq 4$)
or $g=r+8,r+9$ ($r=4$).

######  Proposition 4.1.

  * (i)

$\mathcal{H}^{\mathcal{L}}_{g+r-5,g,r}\neq\emptyset$, $g=r+9$ is irreducible
for every $r\geq 7$.

  * (ii)

$\mathcal{H}^{\mathcal{L}}_{g+r-5,g,r}\neq\emptyset$, $g=r+9$ is reducible for
$r=6$.

  * (iii)

$\mathcal{H}^{\mathcal{L}}_{g+r-5,g,r}\neq\emptyset$, $g=r+9$ is irreducible
for $r=5$.

  * (iv)

$\mathcal{H}^{\mathcal{L}}_{g+r-5,g,r}\neq\emptyset$, $g=r+8$ is irreducible
for $r=4$.

  * (v)

$\mathcal{H}^{\mathcal{L}}_{g+r-5,g,r}\neq\emptyset$, $g=r+9$ is reducible for
$r=4$.

  * (vi)

$\mathcal{H}^{\mathcal{L}}_{g+r-5,g,r}\neq\emptyset$, $g=r+9$ is reducible for
$r=3$.

###### Proof.

  * (i),(ii)

If $r\geq 6$ and $g=r+9$, a curve $C\subset\mathbb{P}^{r}$ of degree
$d=g+r-5=2r+4$ and genus $g$ is an extremal curve lying on a surface of
minimal degree; $\pi(2r+4,r)=r+9$ if $r\geq 6$. A routine computation shows
that if $r\geq 7$ there is only one pair
$(a,b)=(3,7-r)\in\mathbb{N}\times\mathbb{Z}$ satisfying the degree and genus
formula (2), (3) for curves lying on a rational normal surface scroll whereas
there are two such pairs $\\{(a,b)|(3,1),(4,-4)\\}$ if $r=6$. Hence the result
follows by Proposition 2.4; see also [3, Theorem 2.5 and Corollary 2.6, page
122].

  * (iii)

$r=5$: Let $C\subset\mathbb{P}^{5}$ be a smooth curve of genus $g=r+9=14$ and
$d=g+r-5=14$. Note that

$\pi_{1}(14,5)=13\lneq g=14\leq\pi(14,5)=15$

and hence $C$ is a nearly extremal curve lying on a quartic surface
$S\subset\mathbb{P}^{5}$. If $S$ is smooth, $S$ is either a rational normal
scroll or a Veronese surface. If $S$ is a Veronese surface, $C$ is an
isomorphic image a smooth plane curve $C_{0}$ of degree $t$ with $14=d=2t$.
Since $p_{a}(C)=g=14\neq p_{a}(C_{0})=15$, this does not occur. Suppose that
$S$ is a rational normal surface scroll and let $C\in|aH+bL|$. We solve
equations (2) (3) and get $a=3,\frac{9}{2}$. Hence $C\in|3H+2L|$ is trigonal
whose trigonal pencil $g^{1}_{3}$ is cut out by rulings of the scroll. Since

$|K_{S}+C-H|=|(-2H+2L)+C-H|=|4L|$

the very ample and complete hyperplane series $g^{5}_{14}$ of
$C\subset\mathbb{P}^{5}$ is of the form $|K_{C}-4g^{1}_{3}|$. Conversely, on a
trigonal curve $C$ of genus $g\geq 12$, the complete linear series
$|K_{C}-4g^{1}_{3}|$ is very ample by Lemma 2.7. Hence there is an irreducible
family

$\mathcal{G}_{3,0}\subset\widetilde{\mathcal{G}}_{\mathcal{L}}\subset\mathcal{G}^{5}_{14}$

sitting over the family of trigonal curves (and dominating the irreducible
locus $\mathcal{M}^{1}_{g,3}$) whose (general) member is of the form
$|K_{C}-4g^{1}_{3}|$, which induces an embedding into $\mathbb{P}^{5}$.
Therefore we may deduce

$\mathcal{G}_{3,0}\stackrel{{\scriptstyle
bir}}{{\cong}}\dim\mathcal{M}^{1}_{g,3},$
$\dim\mathcal{G}_{3,0}=\dim\mathcal{M}^{1}_{g.3}=2g+1>\lambda(g,g,5)=4g-33,$

and the irreducible family
$\mathcal{H}_{3,0}\subset\mathcal{H}^{\mathcal{L}}_{g,g,5}$ sitting over
$\mathcal{G}_{3,0}$ may constitute a full component.

It is possible that our smooth linearly normal $C\subset\mathbb{P}^{5}$ may
lie on a singular quartic surface, which is a cone over a rational normal
curve in $\mathbb{P}^{4}\subset\mathbb{P}^{5}$. Recalling that smooth curves
on a cone over a rational normal curve are specializations of curves on a
rational normal scroll by Proposition 2.5, we deduce that such curves are in
the the boundary of the component containing the family $\mathcal{H}_{3,0}$
over $\mathcal{G}_{3,0}$.

It finally follows that $\mathcal{H}^{\mathcal{L}}_{14,14,5}$ is irreducible
with only one component whose general element is a trigonal curve lying on a
rational normal scroll in $\mathbb{P}^{5}$ which we specified above.

  * (iv)

$r=4,g=r+8=12$: Since $\pi(11,4)=g=12$, a smooth curve
$C\subset\mathbb{P}^{4}$ with $(d,g)=(11,12)$ is an extremal curve lying on a
cubic surface. By an easy computation there is only one pair
$(a,b)=(4,-1)\in\mathbb{N}\times\mathbb{Z}$ satisfying the degree and genus
formula (2), (3) for curves lying on a rational normal surface scroll. Hence
the irreducibility follows by Proposition 2.4.

  * (v)

$r=4,g=r+9=13,d=g-1=12$: This case is more involved and we need to argue as
follows. First note that $\pi_{1}(12,4)=13$. Hence $C$ lies on a cubic or
quartic surface by [16, Theorem 3.15, page 99]. Suppose $C$ lies on a smooth
cubic surface scroll $S\subset\mathbb{P}^{4}$ and let $C\in|aH+bL|$; cf.
Remark 2.6. We solve equations (2) (3) and get $a=3,\frac{16}{3}$. Hence
$C\in|3H+3L|$ is trigonal whose trigonal pencil $g^{1}_{3}$ is cut out by
rulings of the scroll. Since

$|K_{S}+C-H|=|(-2H+L)+C-H|=|4L|$

the very ample and complete hyperplane series $|H|=g^{4}_{12}$ of
$C\subset\mathbb{P}^{4}$ is of the form $|K_{C}-4g^{1}_{3}|$. By the same
reasoning as in the case (iii), we may deduce that there is an irreducible
family

$\mathcal{G}_{3,0}\subset\widetilde{\mathcal{G}}_{\mathcal{L}}\subset\mathcal{G}^{4}_{14}$

sitting over the family of trigonal curves (and dominating the irreducible
locus $\mathcal{M}^{1}_{g,3}$) whose (general) member is of the form
$|K_{C}-4g^{1}_{3}|$ and induces an embedding into $\mathbb{P}^{4}$. Therefore

$\mathcal{G}_{3,0}\stackrel{{\scriptstyle
bir}}{{\cong}}\dim\mathcal{M}^{1}_{g,3},$
$\dim\mathcal{G}_{3,0}=\dim\mathcal{M}^{1}_{g.3}=2g+1>\lambda(g,g-1,4)=4g-28,$

and the irreducible family
$\mathcal{H}_{3,0}\subset\mathcal{H}^{\mathcal{L}}_{g,g-1,4}$ sitting over
$\mathcal{G}_{3,0}$ has dimension

$\dim\mathcal{H}_{3,0}=\dim\mathcal{M}^{1}_{g,3}+\dim\mathbb{P}GL(5)=51.$

We now suppose that a smooth linearly normal curve $C\subset\mathbb{P}^{4}$ of
degree $d=12$ and genus $g=13$ lies on a quartic surface
$S\subset\mathbb{P}^{4}$, which is one of the following:

    * (1)

An external projection of a quartic scroll or a Veronese surface in
$\mathbb{P}^{5}$,

    * (2)

a del Pezzo surface, possibly with finitely many isolated double points,

    * (3)

a cone over a elliptic quartic curve in $\mathbb{P}^{3}\subset\mathbb{P}^{4}$.

We do not attempt to go through and analyze all the possible cases listed
above, which could well be quite cumbersome. Instead, we choose a reasonable
quartic surface $S\subset\mathbb{P}^{4}$ and get a numerical description of a
smooth curves $C\subset S$ with $(d,g)=(12,13)$. We then proceed to compute
the normal bundle $N_{C|\mathbb{P}^{4}}$, which would eventually justify the
existence of an extra component of $\mathcal{H}^{\mathcal{L}}_{12,13,4}$ other
than the component containing the family $\mathcal{H}_{3,0}$ consisting of
trigonal curves which we already obtained.

Let $S\subset\mathbb{P}^{4}$ be a smooth del Pezzo which is the isomorphic
image of
$\mathbb{P}^{2}_{5}\lhook\joinrel\xrightarrow{|(3;1^{5})|}S\subset\mathbb{P}^{4}$.
Setting $C\in(a;b_{1},\cdots,b_{5})$, we have

$\deg C=3a-\sum b_{i}=12,C^{2}=a^{2}-\sum b_{i}^{2}=2g-2-K_{S}\cdot C=36.$

By Schwartz’s inequality, one has $(\sum b_{i})^{2}\leq 5(\sum b_{i}^{2})$ and
substituting $\sum b_{i}=3a-12$, $\sum b_{i}^{2}=a^{2}-36$ we obtain

$5\,({a}^{2}-36)-\left(12-3\,a\right)^{2}\geq 0\Longleftrightarrow
a=9,b_{1}=b_{2}=\cdots b_{5}=3,$

and therefore

$C\in L:=(9;3^{5}).$

By Riemann-Roch on $S$, Serre duality and Kodaira’s vanishing theorem, one may
easily compute

$h^{0}(S,L)=25.$

Let $\mathcal{H}^{0}_{4}\subset\mathcal{H}_{12,13,4}$ be the irreducible locus
consisting of curves lying on a smooth del Pezzo surface. Recall that a smooth
del Pezzo in $\mathbb{P}^{4}$ is a complete intersection of two quadric
hypersurfaces and is completely determined by a pencil of quadrics in
$\mathbb{P}^{4}$;

$\displaystyle\dim\mathcal{H}^{0}_{4}$
$\displaystyle=\dim\mathbb{G}(1,\mathbb{P}(H^{0}(\mathbb{P}^{4},\mathcal{O}(2))))+\dim\mathbb{P}(H^{0}(S,L))$
(14)
$\displaystyle=\dim\mathbb{G}(1,14)+24=50>\lambda(12,13,4)+\dim\mathbb{P}GL(5)=48.$

The strict inequality in (14) suggests that $\mathcal{H}^{0}_{4}$ (or its
closure) could well be a component other than $\mathcal{H}_{3,0}$. We now
verify that the closure of $\mathcal{H}^{0}_{4}$ is indeed a component of
dimension $50$.

Since

$C\in(9;3^{5})=3(3;1^{5})=3(-K_{S}),$

$C\subset S$ is a divisor cut out on $S$ by a cubic hypersurface and hence $C$
is a complete intersection of two quadrics and a cubic hypersurface. We also
note that

$K_{S}+C\in|(-3;-1^{5})+(9;3^{5})|=|(6;2^{5})|=|2(3;1^{5})|,$

and hence $K_{C}$ is cut out on $C$ by $\mathcal{O}_{S}(2)$;
$\mathcal{O}_{C}(K_{C})\cong\mathcal{O}_{S}(2)_{|C}$. In other words, the
hyperplane series of $C\subset S\subset\mathbb{P}^{4}$ is semi-canonical and

$h^{0}(C,\mathcal{O}_{C}(2))=h^{0}(C,\mathcal{O}_{C}(K_{C}))=g=13.$

We compute the dimension of the tangent space

$T_{C}{\mathcal{H}_{4}}\cong H^{0}(C,N_{C|\mathbb{P}^{4}})$

where $\mathcal{H}_{4}$ is a component containing the irreducible family
$\mathcal{H}^{0}_{4}$. Since $C$ is a complete intersection, the normal bundle
$N_{C|\mathbb{P}^{4}}$ splits;

$N_{C|\mathbb{P}^{4}}=\mathcal{O}_{C}(2)\oplus\mathcal{O}_{C}(2)\oplus\mathcal{O}_{C}(3).$

Noting that $\cal{O}_{C}(3)$ is non-special, we have

$h^{0}(C,N_{C|\mathbb{P}^{4}})=2h^{0}(C,\mathcal{O}_{C}(2))+h^{0}(C,\mathcal{O}_{C}(3))=50=\dim\mathcal{H}^{0}_{4}$

by (14). Hence the irreducible family $\mathcal{H}^{0}_{4}$ is dense in the
component $\mathcal{H}_{4}$. Since
$\dim\mathcal{H}^{0}_{3}=51>\dim\mathcal{H}_{4}$, we have

$\mathcal{H}_{4}\neq\mathcal{H}_{3}$

where $\mathcal{H}_{3}$ is a component containing the irreducible locus
$\mathcal{H}^{0}_{3}$. The reducibility of
$\mathcal{H}^{\mathcal{L}}_{12,13,4}$ readily follows from this.444One may
invoke lower semi-continuity of gonality to ensure that $\mathcal{H}^{0}_{4}$
is not in the boundary of $\mathcal{H}^{0}_{3}$. However without the
computation $\dim H^{0}(C,N_{C|\mathbb{P}^{4}})$, it is presumably possible
that the two irreducible families $\mathcal{H}^{0}_{3}$ and
$\mathcal{H}^{0}_{4}$ may be contained in another component of dimension
strictly bigger than $\dim\mathcal{H}^{0}_{3}$.

  * (vi)

$r=3$, $g=r+9=12$: Let $C\subset\mathbb{P}^{3}$ be a smooth curve with
$(d,g)=(10,12)$. From the exact sequence

$0\rightarrow\mathcal{I}_{C}(3)\rightarrow\mathcal{O}_{\mathbb{P}^{3}}(3)\rightarrow\mathcal{O}_{C}(3)\rightarrow
0$

and Riemann-Roch one computes $H^{0}(\mathbb{P}^{3},\mathcal{I}_{C}(3))\neq
0$, hence $C$ lies on a possibly reducible cubic surface. Indeed there exists
a family $\mathcal{I}_{2}$ consisting of smooth curves of degree $10$ of type
$(3,7)$ on smooth quadrics;

$\dim\mathcal{I}_{2}=\dim\mathbb{P}(H^{0}(\mathbb{P}^{3},\mathcal{O}(2)))+\dim\mathbb{P}(H^{0}(\mathbb{P}^{1}\times\mathbb{P}^{1}\mathcal{O}_{\mathbb{P}^{1}\times\mathbb{P}^{1}}(3,7)))=40.$

There also exists a family $\mathcal{I}_{3}$ consisting of curves on non-
singular cubics, e.g. a general member in the linear system $(9;3^{5},2)$ on
$\mathbb{P}^{2}_{6}\lhook\joinrel\xrightarrow{(3;1^{6})}\mathbb{P}^{3}$;

$\dim\mathcal{I}_{3}=\dim\mathbb{P}(H^{0}(\mathbb{P}^{3},\mathcal{O}(3)))+\dim|(9;3^{5},2)|=40=4\cdot
10.$

Since $\dim\mathcal{I}_{2}=\dim\mathcal{I}_{3}$, one is not in the closure of
the other and hence these two families form two distinct irreducible
components of $\mathcal{H}^{\mathcal{L}}_{g-2,g,3}$ of the same expected
dimension; cf. [23, Theorem 4.1 and Corollary 4.2] for other related results.

∎

######  Remark 4.2.

It is worthwhile to note that
$\mathcal{H}_{g+r-5,g,r}=\mathcal{H}^{\mathcal{L}}_{g+r-5,g,r}$ for triples
$(d,g,r)$ in Proposition 4.1. This follows from the fact that extremal, nearly
extremal or curves with $g=\pi_{1}(d,r)$ are always linearly normal; note that

$\pi(d,r+1)<\pi(d,r+1)+\left[\frac{d}{r}\right]\leq\pi_{1}(d,r)<\pi(d,r).$

######  Remark 4.3.

The following is the current status of our knowledge about the irreducibility
of $\mathcal{H}^{\mathcal{L}}_{d,g,r}$ with low index of speciality
$\alpha=g-d+r$; cf. [5, 22, 23].

  * (i)

If $\alpha=0$, we have $\rho(d,g,r)=g>0$ and there is no reducible
$\mathcal{H}^{\mathcal{L}}_{d,g,r}$. Hence the Modified Assertion of Severi is
trivially true in this case.

  * (ii)

If $\alpha=1$, $\rho(d,g,r)=g-(r+1)$.
$\mathcal{H}^{\mathcal{L}}_{d,g,r}\neq\emptyset$ if and only if $\rho\geq 0$.
Moreover, every non-empty $\mathcal{H}^{\mathcal{L}}_{d,g,r}$ is irreducible
and therefore Modified Assertion of Severi also holds in this case.

  * (iii)

If $\alpha=2$, $\rho(d,g,r)=g-2(r+1)$. One has
$\mathcal{H}^{\mathcal{L}}_{d,g,r}\neq\emptyset$ if and only if $g\geq r+3$
and every non-empty $\mathcal{H}^{\mathcal{L}}_{d,g,r}$ is irreducible, which
follows from the irreducibility of Hurwitz scheme. Hence the Modified
Assertion of Severi holds everywhere beyond the Brill-Noether range.

  * (iv)

If $\alpha=3$, $\rho(d,g,r)=g-3(r+1)$. It is known that if $g\geq 2r+3$,
$\mathcal{H}^{\mathcal{L}}_{d,g,r}$ is irreducible, which basically follows
from the irreducibility of Severi variety of plane nodal curves; cf. [5].
Therefore the Modified Assertion of Severi holds far beyond the Brill-Noether
range. The lower bound $g\geq 2r+3$ for the irreducible
$\mathcal{H}^{\mathcal{L}}_{g+r-3,g,r}$ is sharp for every $r\geq 5$. Indeed
for all the values $g$ in the range $r+5\leq g\leq 2r+2$ such that $r\geq 5$,
$\mathcal{H}^{\mathcal{L}}_{d,g,r}$ is reducible. Moreover all the triples
$(d,g,r)=(g+r-3,g,r),r\geq 3$ for which $\mathcal{H}^{\mathcal{L}}_{d,g,r}$ is
irreducible have been idetified; cf. [22, Theorem 1.1, Remark 3.1, Theorem
3.9, Corollary 3.10] and references therein.

  * (v)

For $\alpha=4$, our overall knowledge on the irreducibility of
$\mathcal{H}^{\mathcal{L}}_{d,g,r}$ drastically decreases compared with the
Hilbert schemes having smaller index of speciality. Unless the genus $g$ is
near to the minimal possible value with respect to $r$, virtually nothing is
known about the irreducibility of the $\mathcal{H}^{\mathcal{L}}_{d,g,r}$
either inside or outside the Brill Noether range.

  * (vi)

In the next section, we proceed one step further to determine the
irreducibility of $\mathcal{H}^{\mathcal{L}}_{d,g,r}$ with $\alpha=5$ when the
genus $g$ is very near to minimal possible value with respect to $r$.

##  5\. Irreducibility of $\mathcal{H}^{\mathcal{L}}_{2r+5,r+10,r}$

In Theorem 3.2, we saw that if $g=r+10$,
$\mathcal{H}^{\mathcal{L}}_{g+r-5,g,r}\neq\emptyset$ if and only if $3\leq
r\leq 8$. However when $g=r+9$,
$\mathcal{H}^{\mathcal{L}}_{g+r-5,g,r}\neq\emptyset$ for every $r\geq 3$. A
similar phenomenon of this kind occurs for low index of speciality
$3\leq\alpha\leq 4$; cf. [22, Proposition 3.3] and [23, Theorem 2.4]. In this
section we determine the irreducibility of this peculiar Hilbert scheme
$\mathcal{H}^{\mathcal{L}}_{2r+5,r+10,r}$ for every $r\neq 4$. We start to
prove the following.

######  Theorem 5.1.

$\mathcal{H}^{\mathcal{L}}_{2r+5,r+10,r}$ is irreducible if $r=6,7$ and
reducible if $r=8$.

###### Proof.

Given a component
$\mathcal{G}\subset\widetilde{\mathcal{G}}_{\mathcal{L}}\subset\mathcal{G}^{r}_{g+r-5}$
whose general element $(p,\mathcal{D})\in\cal{G}$ corresponds to a very ample
and complete linear series $\mathcal{D}$ on the curve $C=\xi^{-1}(p)$, set
$\mathcal{E}=\mathcal{D}^{\vee}=g^{4}_{13}$. Note that
$\mathcal{E}=g^{4}_{13}$ is not compounded; otherwise we may use Lemma 2.10 to
conclude that $\mathcal{D}$ is not very ample. We also note that $\mathcal{E}$
is base-point-free since $\pi(12,4)=15<r+10=g$. In the range $6\leq r\leq 8$,

$\pi_{1}(13,4)=15<g=r+10\leq\pi(13,4)=18,$

hence a curve of (geometric) genus $g=r+10$ and degree $13$ in
$\mathbb{P}^{4}$ lies on a surface of minimal degree in $\mathbb{P}^{4}$; cf.
[16, Theorem 3.15, page 99]. Let $C_{\mathcal{E}}\subset\mathbb{P}^{4}$ be the
image curve of the morphism induced by $\mathcal{E}$ and let
$S\subset\mathbb{P}^{4}$ be the cubic surface containing $C_{\mathcal{E}}$.

(A) We first assume that $S$ is a non-singular cubic rational normal surface
scroll. Recall that there are obvious isomorphisms

$S\cong\mathbb{F}_{1}\cong\mathbb{P}^{2}_{1}$

together with natural correspondences between generators of Picard groups;

$\displaystyle\textrm{Pic}S$ $\displaystyle\cong$
$\displaystyle\textrm{Pic}(\mathbb{F}_{1})$ $\displaystyle\cong$
$\displaystyle\textrm{Pic}(\mathbb{P}^{2}_{1})$ $\displaystyle(H,L)$
$\displaystyle\longleftrightarrow$ $\displaystyle(e+2f,f)$
$\displaystyle\longleftrightarrow$ $\displaystyle(2l-\tilde{e},l-\tilde{e}),$
(15)

$e\in\textrm{Pic}(\mathbb{F}_{1})$ is the class of the section with minimal
self intersection $e^{2}=-1$, $f$ the class of fibre of the Hirzebruch surface
$\mathbb{F}_{1}\rightarrow\mathbb{P}^{1}$; $l$ the class of a line in
$\mathbb{P}^{2}_{1}$, $\widetilde{e}$ the class of the exceptional divisor on
$\mathbb{P}^{2}_{1}$. By abusing notation we shall make no distinction between
$e\in\textrm{Pic}(\mathbb{F}_{1})$ and
$\widetilde{e}\in\textrm{Pic}(\mathbb{P}^{2}_{1})$ and use the same symbol
$e$.

Let $C_{\mathcal{E}}\in|aH+bL|$ and we solve equations (2) and (3);
$n=3,d=13,~{}p_{a}(C_{\mathcal{E}})=r+10$. The solutions of integer pairs
$(a,b)\in\mathbb{N}\times\mathbb{Z}$ exist only when $r=8$;
$(a,b)\in\\{(5,-2),(4,1)\\}$. Hence $C_{\mathcal{E}}$ is smooth only if $r=8$
and is singular if $r=6,7$. Set $\cal{N}_{1}:=\mathcal{O}_{S}(5H-2L)$ and
$\cal{N}_{2}:=\mathcal{O}_{S}(4H+L)$.

(A-1): We first deal with the case $r=8$. By the correspondence (5), we have

  * (i)

$C_{\mathcal{E}}\in|\cal{N}_{1}|=|5H-2L|=|5(e+2f)-2f|=|8f+5e|=|8l-3e|=(8;3)$

$|D|:=|K_{S}+C_{\mathcal{E}}-H|=|2H-L|=|2(e+2f)-f|=|3l-e|=(3;1)$

$\mathcal{E}^{\vee}=|K_{C_{\mathcal{E}}}-\mathcal{E}|=(3;1)_{|{C_{\mathcal{E}}}}$

  * (ii)

$C_{\mathcal{E}}\in|\cal{N}_{2}|=|4H+L|=|4(e+2f)+f|=|9f+4e|=|9l-5e|=(9;5)$

$|D|:=|K_{S}+C_{\mathcal{E}}-H|=|H+2L|=|e+4f|=|4l-3e|=(4;3)$

$\mathcal{E}^{\vee}=|K_{C_{\mathcal{E}}}-\mathcal{E}|=(4;3)_{|{C_{\mathcal{E}}}}$

Let $\rho_{\cal{N}_{i}}:H^{0}(S,\mathcal{O}_{S}(D))\rightarrow
H^{0}(C_{\mathcal{E}},\mathcal{O}_{S}(D)\otimes\mathcal{O}_{C_{\mathcal{E}}})$
be the restriction map. We have

$\ker\rho_{\cal{N}_{i}}\cong
H^{0}(S,\cal{N}_{i}^{-1}\otimes\mathcal{O}_{S}(D))\cong
H^{0}(S,\mathcal{O}_{S}(-3H+L))=0,$

$H^{1}(S,\mathcal{O}_{S}(-3H+L))=0$ by Kodaira vanishing theorem since
$|3H-L|$ is (very) ample. Hence $\rho_{\cal{N}_{i}}$ ($i=1,2$) is an
isomorphism. Hence in both cases (i) and (ii), the linear system
$\mathcal{E}^{\vee}$ on $C_{\mathcal{E}}$ is very ample since the
corresponding linear system $(3;1)$ or $(4;3)$ on $S\cong\mathbb{P}^{2}_{1}$
is very ample. One may interpret our current situation as follows.

Observation 1: Given a very ample
$(\mathcal{D}=g^{r}_{d},\widetilde{C})\in\mathcal{G}\subset\widetilde{\mathcal{G}}_{\mathcal{L}}\subset\mathcal{G}^{r}_{d}$,
we take the residual series $\mathcal{D}^{\vee}=\mathcal{E}$ on
$\widetilde{C}$. If $r=8$, $\mathcal{E}$ induces a morphism onto a smooth
curve $C_{\mathcal{E}}\subset\mathbb{P}^{4}$ sitting on the surface $S$ of
minimal degree belonging to one of the linear systems

$\mathcal{N}_{1}=|5H-2L|\textrm{ or }\mathcal{N}_{2}=|4H+L|.$

Therefore $C_{\mathcal{E}}$ may be regarded as an element of the (set
theoretic) union of Severi varieties
$\Sigma_{\mathcal{N}_{1},\delta}\cup\Sigma_{\mathcal{N}_{2},\delta}\subset\textrm{Div}S$
where $\delta=0$.

One may do the same thing for the cases $r=6,7$. Upon fixing a rational normal
surface scroll $S\subset\mathbb{P}^{4}$ once and for all (up to
$\textrm{Aut}(S)$) and putting $\delta=8-r$, we have a natural generically
injective correspondence

$\displaystyle\widetilde{\mathcal{G}}_{\mathcal{L}}\hskip
24.0pt\stackrel{{\scriptstyle\tau}}{{\dashrightarrow}}$
$\displaystyle\widetilde{\mathcal{G}}^{\vee}_{\mathcal{L}}\hskip
24.0pt\stackrel{{\scriptstyle\iota}}{{\dashrightarrow}}$
$\displaystyle{\tilde{\Sigma}}_{\mathcal{N}_{1},\delta}\cup{\tilde{\Sigma}}_{\mathcal{N}_{2},\delta}\subset\textrm{Div}{S}/\textrm{Aut}(S)$
$\displaystyle\mathcal{D}\hskip 24.0pt\longmapsto$
$\displaystyle\mathcal{D}^{\vee}\hskip 24.0pt\longmapsto$ $\displaystyle\hskip
20.0ptC_{\mathcal{D}^{\vee}}=C_{\mathcal{E}}$

where ${\tilde{\Sigma}}_{\mathcal{N}_{i},\delta}$ is the image of
$\Sigma_{\mathcal{N}_{i},\delta}$ under the projection
$\textrm{Div}S\rightarrow\textrm{Div}S/\textrm{Aut}(S)$. Let
$\mathcal{U}\subset{\tilde{\Sigma}}_{\mathcal{N}_{1},\delta}\cup{\tilde{\Sigma}}_{{\mathcal{N}_{2}},\delta}$
be an open set on which the inverse of the rational map $\iota\circ\tau$ is
defined.

In the case $r=8$, as we saw in (A-1), for a general
$C_{\mathcal{E}}\in\mathcal{N}_{i}=\Sigma_{\mathcal{N}_{i},0}$,
$|K_{S}+C_{\mathcal{E}}-\mathcal{O}_{S}(1)|_{C_{\mathcal{E}}}$ is very ample,
thus we may conclude

$\mathcal{U}\cap{\tilde{\Sigma}}_{\mathcal{N}_{i},\delta}\neq\emptyset\textrm{
for }i=1,2.$

Hence there are two irreducible families
$\mathcal{G}_{i}\subset\widetilde{\mathcal{G}}_{\mathcal{L}}$ corresponding to
${\tilde{\Sigma}}_{\mathcal{N}_{i},\delta}$ of dimensions 555In general
$\textrm{Aut}(\mathbb{F}_{n})$ is isomorphic to the semi-direct product
$\textrm{Sym}^{n}\mathbb{C}^{2}\ltimes GL(2)/\mu_{n}$ where $\mu_{n}$ is the
group of $n$-th roots of unity in the group of scalar matrices; cf. [8, page
12]. Hence $\textrm{Aut}(\mathbb{F}_{1})\cong\mathbb{C}^{2}\ltimes GL(2)$.
Alternatively, by recalling that rational normal surface scrolls in
$\mathbb{P}^{s}$ move in an irreducible family $\mathcal{S}_{s}$ of dimension
$(s+3)(s-1)-3$,
$\dim\mathcal{G}_{i}=\dim\mathcal{N}_{i}+\dim\mathcal{S}_{4}-\dim\mathbb{P}(GL(5))=\dim\mathcal{N}_{i}-6.$

$\dim\mathcal{G}_{i}=\dim\mathcal{N}_{i}/\textrm{Aut}(S)=\dim\mathcal{N}_{i}-\dim(\mathbb{C}^{2}\ltimes
GL(2))=\dim\mathcal{N}_{i}-6.$

Therefore it follows that $\widetilde{\mathcal{G}}_{\mathcal{L}}$ which
$\mathcal{H}^{\mathcal{L}}_{21,18.8}$ sits over as a
$\operatorname{Aut}(\mathbb{P}^{8})$-bundle is reducible with two components
$\mathcal{G}_{i}$ corresponding to two irreducible Severi varieitis
$\Sigma_{\mathcal{N}_{i},0}$, $i=1,2$. We remark that the one of the two
irreducible families $\mathcal{G}_{i}$ is not in the boundary of the other by
semi-continuity of gonality;

$\dim\mathcal{G}_{2}=33>\dim\mathcal{G}_{1}=32\mbox{ whereas
}\textrm{gon}(C_{1})>\textrm{gon}(C_{2})\mbox{ for }C_{i}\in\mathcal{N}_{i}$

Observation 2: We remark that the original smooth curve
$C\subset\mathbb{P}^{8}$ with $(d,g)=(21,18)$ is neither extremal nor nearly
extremal; $\pi_{1}(d,r)=g$. Therefore it might be quite complicated to carry
out comprehensive analysis of curves in $\mathbb{P}^{8}$ lying on a surface of
degree eight or seven. However the residual curve
$C_{\mathcal{E}}\subset\mathbb{P}^{4}$ is an extremal curve lying on a surface
of minimal degree, from which we were able to deduce the reducibility
$\mathcal{H}^{\mathcal{L}}_{21,18.8}$ from the reducibility of the family of
residual curves which are members of two typical Severi varieties
$\Sigma_{\mathcal{N}_{i},0}(i=1,2)$ on the rational surface $S$.

(A-2): In the same vein, we proceed to handle the cases $6\leq r\leq 7$. In
this case, the residual (singular) curve
$C_{\mathcal{E}}\subset\mathbb{P}^{4}$ is not an extremal curve but a nearly
extremal curve;

$\pi_{1}(13,4)=15\lneq g=r+10\lneq\pi(13,4)=18.$

By Remark 2.2 (i), we may assume that $C_{\mathcal{E}}$ is a nodal curve lying
on a surface of minimal degree $S\cong\mathbb{F}_{1}$ with $\delta=8-r$ nodes
as its only singularities. Again, the residual curve $C_{\mathcal{E}}$ – as a
divisor of $S\cong\mathbb{F}_{1}$ – may be regarded as an element in the union
of Severi varieties

$\Sigma_{\mathcal{N}_{1},8-r}\cup\Sigma_{\mathcal{N}_{2},8-r}.$

We now take a curve $C\subset S\subset\mathbb{P}^{4}$ corresponding to a
general element of $\Sigma_{\mathcal{N}_{i},8-r}$. We blow up
$\mathbb{F}_{1}\cong\mathbb{P}^{2}_{1}$ at the $(8-r)$ nodal points to get the
surface $\mathbb{P}^{2}_{9-r}$. We may assume that the singularities of $C$
are not in the exceptional divisor $e$ of $\mathbb{P}^{2}_{1}$. Let
$\widetilde{C}$ be the proper(strict) transformation of $C$ under the blow up
$\mathbb{P}^{2}_{9-r}\stackrel{{\scriptstyle\pi}}{{\rightarrow}}\mathbb{P}^{1}_{1}\cong\mathbb{F}_{1}\cong
S$. Let $e$, $e_{1},\cdots e_{8-r}$ be the exceptional divisors of the blown
up surface $\mathbb{P}^{2}_{9-r}$ and set
$\widetilde{H}=\pi^{*}\mathcal{O}_{S}(1)$. We will determine the class of
$\widetilde{C}$ and the linear system
$|K_{\mathbb{P}^{2}_{9-r}}+\widetilde{C}-\widetilde{H}|$ which may cut
$\cal{E}^{\vee}$ on $\widetilde{C}$. Our goal is to check if

$\mathcal{D}=|K_{\mathbb{P}^{2}_{9-r}}+\widetilde{C}-\widetilde{H}|_{\widetilde{C}}$

is very ample. In other words, we would like to examine if the series
$\mathcal{D}$ on smooth $\widetilde{C}$ – which is the residual series of the
series cut out by hyperplanes in $\mathbb{P}^{4}$ – is very ample. If this is
the case, we may conclude that either one of the Severi varieties
$\Sigma_{\mathcal{N}_{i},{8-r}}$ – which is irreducible by a work of Tyomkin
[34] – corresponds to an irreducible component of
$\widetilde{\mathcal{G}}_{\mathcal{L}}\subset\mathcal{G}^{r}_{2r+5}$. We
denote by

$\Sigma_{\mathcal{N}_{i},\delta}^{\vee}:=\\{|K_{\mathbb{P}^{2}_{9-r}}+\widetilde{C}-\widetilde{H}|_{|{\widetilde{C}}}:C\in\Sigma_{\mathcal{N}_{i},\delta},\,\widetilde{H}=\pi^{*}\mathcal{O}_{S}(1),\,\mathbb{P}^{2}_{9-r}\stackrel{{\scriptstyle\pi}}{{\rightarrow}}S\\}$
(16)

the family consisting of such $\mathcal{D}$’s.

  * (iii)

$C\in\Sigma_{\mathcal{N}_{1},{8-r}}\subset\mathcal{N}_{1}=|5H-2L|$: With a
minor modification of the previous computation in (A-1)-(i), we have

$\widetilde{C}\in(8;3,2^{8-r})$

and may deduce that

$\displaystyle|D|:$
$\displaystyle=|K_{\mathbb{P}^{2}_{9-r}}+\widetilde{C}-\widetilde{H}|$
$\displaystyle=|(-3;-1^{9-r})+(8;3,2^{8-r})-(2;1,0^{8-r})|=(3;1^{9-r}),$

which is very ample inducing an embedding
$\varphi:\mathbb{P}^{2}_{9-r}\stackrel{{\scriptstyle(3;1^{9-r})}}{{\longrightarrow}}\mathbb{P}^{r}$
as well as an isomorphism
$\widetilde{C}\cong\varphi({\widetilde{C}})\subset\mathbb{P}^{r}$;

$\deg\varphi({\widetilde{C}})=\widetilde{C}\cdot(K_{\mathbb{P}^{2}_{9-r}}+\widetilde{C}-\widetilde{H})=2r+5,~{}g(\varphi(\widetilde{C}))=g(\widetilde{C})=r+10.$

Borrowing the notation used in the proof of Theorem 3.2 (7), we note that the
linear system

$(8;3,2^{8-r})=\mathcal{L}(0,r)$

to which $\widetilde{C}$ belongs, is a special case of the one we encountered
there. Furthermore, one may check that the restriction map

$\rho_{\mathcal{L}(0,r)}:H^{0}(\mathbb{P}^{2}_{9-r},\mathcal{O}_{\mathbb{P}^{2}_{9-r}}(D))\rightarrow
H^{0}(\widetilde{C},\mathcal{O}_{\mathbb{P}^{2}_{9-r}}(D)\otimes\mathcal{O}_{\widetilde{C}})$

is an isomorphism;

$\displaystyle\ker\rho_{\mathcal{L}(0,r)}$
$\displaystyle=H^{0}(\mathbb{P}^{2}_{9-r},\mathcal{O}_{\mathbb{P}^{2}_{9-r}}(D)\otimes{\mathcal{L}(0,r)}^{-1})$
$\displaystyle=H^{0}(\mathbb{P}^{2}_{9-r},\mathcal{O}_{\mathbb{P}^{2}_{9-r}}(K_{\mathbb{P}^{2}_{9-r}}-\widetilde{H}))$
$\displaystyle=H^{0}(\mathbb{P}^{2}_{9-r},\mathcal{O}_{\mathbb{P}^{2}_{9-r}}(-(5;2,1^{8-r})))=0$

and
$H^{1}(\mathbb{P}^{2}_{9-r},\mathcal{O}_{\mathbb{P}^{2}_{9-r}}(D)\otimes{\mathcal{L}(0,r)}^{-1})=0$
by Kodaira vanishing theorem. Therefore a general element of
$\Sigma_{\mathcal{N}_{1},\delta}^{\vee}$ is very ample and
$\Sigma_{\mathcal{N}_{1},\delta}^{\vee}$ corresponds to a component of
$\widetilde{\mathcal{G}}_{\mathcal{L}}\subset\mathcal{G}^{r}_{2r+5}$.

  * (iv)

$C\in\Sigma_{\mathcal{N}_{2},{8-r}}\subset\mathcal{N}_{2}=|4H-L|$: In the same
way, we get

$\widetilde{C}\in(9,5,2^{8-r})$

and

$\displaystyle|D|:=|K_{\mathbb{P}^{2}_{9-r}}+\widetilde{C}-\widetilde{H}|$
$\displaystyle=(-3;-1^{9-r})+(9;5,2^{8-r})-(2,1,0^{8-r})$
$\displaystyle=(4;3,1^{8-r}),$

which is not very ample unless $r=8$ by [13]. On the other hand, the
corresponding restriction map is still an isomorphism. Finally we remark that
the $(-1)$ curve $l-e-e_{1}=(1;1,1,0^{7-r})$ is contracted by the morphism
induced by $(4;3,1^{8-r})$, whereas $\widetilde{C}\cdot(l-e-e_{1})=2$ hence
producing a singularity on the image curve.

(B) We assume that $C_{\mathcal{E}}$ lies on a rational normal cone
$S\subset\mathbb{P}^{4}$. We borrow the notation from Remark 2.3 (iii) and
take $n=3$, $d=13$ there. Let $\widetilde{C}_{\cal{E}}$ be the proper
transformation of $C_{\mathcal{E}}$ under the minimal desingularization
$\mathbb{F}_{3}\xrightarrow{\ \ }S\subset\mathbb{P}^{4}$ given by
$|C_{0}+3f|$. By (5) and (4), we have

$(k,m,p_{a}(\widetilde{C}{E}))=\begin{cases}(4,1,18)\\\ (3,4,15)\\\
(2,7,9)\end{cases}$

Note that the second and third triples are not possible since
$p_{a}(\widetilde{C}{{E}})<g=r+10=g(\widetilde{C}_{\cal{E}})$, a
contradiction. Hence we only have $(k,m,p_{a}(\widetilde{C}{E}))=(4,1,18)$.
(B-1): If $r=8$, $C_{\mathcal{E}}\subset S$ is smooth:
$p_{a}(\widetilde{C}{{E}})=18=g$. By Propostion 2.5, $C_{\mathcal{E}}$ is a
specialization of curves lying on a smooth rational normal scroll. (B-2): If
$r=7$, $\widetilde{C}{{E}}$ is singular with a double point
$p_{0}\in\widetilde{C}{E}$. Setting $\widetilde{C}_{\cal{E}}\in|aC_{0}+bf|$,
we have $\widetilde{C}_{\cal{E}}\cdot(C_{0}+3f)=b=13,a=k=4$ and
$\widetilde{C}_{\cal{E}}\in|4C_{0}+13f|$. Choose $f_{0}$, the fibre containing
the unique singular point $p_{0}\in\widetilde{C}{E}$. Note that
$(k,m,p_{a}(\widetilde{C}{E}))=(4,1,18)$, thus $C_{\mathcal{E}}$ is smooth at
the vertex of the cone $S\subset\mathbb{P}^{4}$ and we may assume $p_{0}\notin
C_{0}$. We blow up $\mathbb{F}_{3}$ at $p_{0}\in\widetilde{C}_{\cal{E}}$ and
let $e$ be the exceptional divisor of the blow up
$\mathbb{F}_{3,1}\stackrel{{\scriptstyle\pi}}{{\rightarrow}}\mathbb{F}_{3}$.
Let $\tilde{f}_{0}$ ($\widehat{C}$ resp.) be the proper transformation of
$f_{0}$ ($\widetilde{C}_{\cal{E}}$ resp.). By abusing notation, we denote
$\pi^{*}(C_{0})~{}~{}\&~{}~{}\pi^{*}(f)$ by $C_{0}~{}~{}\&~{}~{}f$. We have

$\widehat{C}\in|4C_{0}+13f-2e|$

and consider

$\displaystyle\mathcal{M}$
$\displaystyle:=|\widehat{C}+K_{\mathbb{F}_{3,1}}-(C_{0}+3f)|$
$\displaystyle=|(4C_{0}+13f-2e)+(-2C_{0}-5f+e)-(C_{0}+3f)|$
$\displaystyle=|C_{0}+5f-e|.$

Since
$\mathcal{O}_{\mathbb{F}_{3,1}}(\mathcal{M}-\widehat{C})=\mathcal{O}_{\mathbb{F}_{3,1}}(-(3C_{0}+8f-e))$
and $f\cdot(\mathcal{M}-\widehat{C})<0$, we see that
$h^{0}({\mathbb{F}_{3,1}},\mathcal{O}(\mathcal{M}-\widehat{C}))=0$ implying
the injectivity of the restriction map

$\rho:H^{0}({\mathbb{F}_{3,1}},\mathcal{O}(\mathcal{M}))\longrightarrow
H^{0}(\widehat{C},\mathcal{O}(\mathcal{M})\otimes\mathcal{O}_{\widehat{C}}).$

Suppose $\rho$ is not surjective, i.e. assume

$\mathrm{Im}(\rho)\subsetneq
H^{0}(\widehat{C},\mathcal{M}\otimes\mathcal{O}_{\widehat{C}}).$

By the projection formula,
$\pi_{*}\pi^{*}\mathcal{O}_{\mathbb{F}_{3}}(h+5f)=\mathcal{O}_{\mathbb{F}_{3}}(h+5f)$

$\displaystyle
h^{0}(\mathbb{F}_{3,1},\pi^{*}\mathcal{O}_{\mathbb{F}_{3}}(h+5f))$
$\displaystyle=h^{0}(\mathbb{F}_{3},\pi_{*}\pi^{*}\mathcal{O}_{\mathbb{F}_{3}}(h+5f)))$
$\displaystyle=h^{0}(\mathbb{F}_{3},\mathcal{O}(h+5f))=9$

and hence

$\displaystyle h^{0}(\mathbb{F}_{3,1},\mathcal{O}(\mathcal{M}))$
$\displaystyle=h^{0}(\mathbb{F}_{3,1},\mathcal{O}(h+5f-e))$
$\displaystyle=h^{0}(\mathbb{F}_{3},\mathcal{O}(h+5f))-1=8.$

Then by the injectivity and the non-surjectivity of $\rho$,

$\displaystyle
r=\dim\mathbb{P}(H^{0}(\widehat{C},\mathcal{O}(\mathcal{M})\otimes\mathcal{O}_{\widehat{C}}))$
$\displaystyle>\dim\mathbb{P}(H^{0}(\mathbb{F}_{3,1},\mathcal{O}(\mathcal{M}))$
$\displaystyle=\dim\mathbb{P}(\mathrm{Im}(\rho))=7.$

Since
$|\mathrm{Im}(\rho)|\subsetneq\mathbb{P}(H^{0}(\widehat{C},\mathcal{O}(\mathcal{M})\otimes\mathcal{O}_{\widehat{C}}))$
induces a morphism birational onto its image, the complete linear system
$\mathbb{P}(H^{0}(\widehat{C},\mathcal{O}(\mathcal{M})\otimes\mathcal{O}_{\widehat{C}}))$
is still birationally very ample, which contradicts the Castelnuovo genus
bound for a curve of degree
$\widehat{C}\cdot\mathcal{M}=(4h+13f-2e)\cdot(h+5f-e)=19$ in
$\mathbb{P}^{r\geq 8}$; $\pi(19,8)=15<g=16$. Therefore we have

$\mathrm{Im}(\rho)=H^{0}(\widehat{C},\mathcal{O}(\mathcal{M})\otimes\mathcal{O}_{\widehat{C}})$

and the restriction map $\rho$ is an isomorphism. Note that

$\tilde{f}_{0}^{2}=-1,\tilde{f}_{0}\cdot
e=1,C_{0}\cdot\tilde{f}_{0}=1,\widehat{C}\cdot\tilde{f}_{0}=2,\mathcal{M}\cdot\tilde{f}_{0}=(C_{0}+5f-e)\cdot\tilde{f}_{0}=0.$

Hence the morphism $\psi$ induced by
$\mathcal{M}=|\widehat{C}+K_{\mathbb{F}_{3,1}}-(h+3f)|$ contracts the $(-1)$
curve $\tilde{f}_{0}$ and the image $\psi(\widehat{C})\subset\mathbb{P}^{7}$
acquires a singularity since $\widehat{C}\cdot\tilde{f}_{0}=2$. Recall that
$\cal{E}=\mathcal{O}_{\widehat{C}}(1)$. Hence

$\mathbb{P}(H^{0}(\widehat{C},\mathcal{O}(\mathcal{M})\otimes\mathcal{O}_{\widehat{C}}))=|\cal{E}^{\vee}|=|\cal{D}|$

is not very ample, a contradiction.

(B-3): If $r=6$, $\widetilde{C}{{E}}\subset\mathbb{F}_{3}$ is singular. We
adopt the same strategy (as for the case $r=7$) to show that a general element
in the Severi variety of curves of geometric genus $g=16$ on $\mathbb{F}_{3}$
in $\mathcal{L}=|4h+13f|$ does not have very ample $|K_{\widehat{C}}(-1)|$,
where $\widehat{C}$ is the normalization of
$\widetilde{C}_{\cal{E}}\subset\mathbb{F}_{3}$. Since being very ample is an
open condition, we may assume that $\widetilde{C}_{\cal{E}}\subset\cal{F}_{3}$
has two nodes which lie outside the section $C_{0}\subset\mathbb{F}_{3}$ of
minimal self intersection. Therefore all the argument remain parallel to the
case $r=7$ – except blowing up $\mathbb{F}_{3}$ at two nodal points – and we
omit details.

Conclusion:

  1. (1)

From (A-2), a general element of the family
$\Sigma_{\mathcal{N}_{2},\delta}^{\vee}$ is not very ample if $\delta=8-r\geq
1$. Thus we have

$\mathcal{U}\cap\Sigma_{\mathcal{N}_{2},\delta}=\emptyset\textrm{ and
}\mathcal{U}\cap\Sigma_{\mathcal{N}_{1},\delta}\neq\emptyset$

  2. (2)

From (B), the family of nodal curves lying on a rational normal cone $S$ does
not have very ample residual hyperplane series and hence does contribute to a
component of $\mathcal{H}^{\mathcal{L}}_{2r+5,r+10,r}$ when $r=6,7$.

  3. (3)

The irreducibility of $\mathcal{H}^{\mathcal{L}}_{2r+5,r+10,r}$ for $r=6,7$
readily follows.

∎

######  Remark 5.2.

(a) We saw in the course of the proof of Theorem 5.1 that the peculiar and
seemingly artificial linear system (7) on del Pezzo surfaces which we used in
the proof of Theorem 3.2 arises naturally by analyzing nearly extremal curves
induced by $\mathcal{E}=\mathcal{D}^{\vee}$ corresponding to a general element
of a component of $\mathcal{H}^{\mathcal{L}}_{2r+5,r+10,r}$. Another such
linear system (10) in Theorem 3.2) corresponds to a general element of a
component of $\mathcal{H}^{\mathcal{L}}_{2r+6,r+11,r}$ – another peculiar
Hilbert scheme which is non-empty only for low $r$ – and this
$\mathcal{H}^{\mathcal{L}}_{2r+6,r+11,r}$ will be studied in the next section.

(b) The generic example $\mathcal{L}(0,r)$ (for $r=6,7,8$) corresponding to
$\Sigma_{\mathcal{N}_{1},8-r}$ remain very ample for $3\leq r\leq 5$, .
However, in the cases $3\leq r\leq 5$, it is possible that $C_{\mathcal{E}}$
may lie on a surface in $\mathbb{P}^{4}$ of degree $t\geq 4$. Therefore one
cannot determine the irreducibility of
$\mathcal{H}^{\mathcal{L}}_{2r+5,r+10,r}$ solely based on the irreducibility
of the corresponding Severi varieties.

######  Remark 5.3.

  * (i)

The irreducibility of $\mathcal{H}^{\mathcal{L}}_{2r+5,r+5,r}$ for $r=5$ is
known by a recent work of the author jointly with E. Ballico [6, Theorem 4.1].

  * (ii)

For $r=4$, the irreducibility of $\mathcal{H}^{\mathcal{L}}_{2r+5,r+10,r}$ has
not been settled yet. However, we are able to show that
$\mathcal{H}^{\mathcal{L}}_{13,14,4}$ has a generically reduced component of
minimal possible dimension as follows.

######  Proposition 5.4.

$\cal{H}_{13,14,4}$ has a component of the expected dimension

$\cal{X}(13,14,4)=\lambda(13,14,4)+\dim\operatorname{Aut}(\mathbb{P}^{4})=52.$

###### Proof.

We borrow notations and computations from the proof of Theorem 3.2 (c). Take a
general $C\in(8;3,2^{4})=\cal{L}(0,4)$; cf. (7). We have
$C\subset\mathbb{P}^{2}_{5}\subset\mathbb{P}^{4}$ and $(d,g)=(13,14)$. We
consider the irreducible family $\cal{F}$ of smooth curves lying on smooth del
Pezzo surfaces in $\mathbb{P}^{4}$;

$\cal{F}:=\\{C\,|\,(d,g)=(13,14),\,C\subset
T\cong\mathbb{P}^{2}_{5}\subset\mathbb{P}^{4},\,C\in(8;3,2^{4})\\}.$

Note that the residual curve (by definition, the curve induced by the residual
series of the hyperplane series of $C$) $C^{\vee}\subset\mathbb{P}^{4}$ lies
on a smooth cubic scroll $S\subset\mathbb{P}^{4}$,
$C^{\vee}\in|5H-2L|=\cal{N}_{1}$ and $C^{\vee}\in\Sigma_{\cal{N}_{1,4}}$; cf.
part (A-2) (iii) of the proof of Theorem 5.1. We have
$\dim\Sigma_{\cal{N}_{1,4}}=\dim|\cal{N}_{1}|-4=34$, hence

$\dim\cal{F}=\dim\Sigma_{\cal{N}_{1,4}}-\dim\operatorname{Aut}(S)+\dim\operatorname{Aut}(\mathbb{P}^{4})=34-6+24=52.$

We want to estimate the dimension of a component
$\mathcal{H}\subset\mathcal{H}_{13,14,4}$ containing $\cal{F}$. To do this, we
compute $h^{0}(C,N_{C|\mathbb{P}^{4}})=\dim T_{C}{\mathcal{H}}$.

We consider the standard exact sequence

$0\rightarrow N_{C|S}\rightarrow N_{C|\mathbb{P}^{4}}\rightarrow
N_{{S|\mathbb{P}^{4}}_{|C}}\rightarrow 0.$

We note that

$N_{C|S}=\mathcal{O}_{S}(C)_{|C}=\mathcal{O}_{S}(C)\otimes\mathcal{O}_{C}.$

(i) Claim: $h^{0}(C,N_{C|S})=26$ and $h^{1}(C,N_{C|S})=0$. This follows from
the exact sequence

$0\rightarrow\mathcal{O}_{S}\rightarrow\mathcal{O}_{S}(C)\rightarrow\mathcal{O}_{S}(C)\otimes\mathcal{O}_{C}\rightarrow
0;$

a routine computation yields

$h^{0}(S,\mathcal{O}_{S}(C))=\frac{(8+1)(8+2)}{2}-\frac{3(3+1)}{2}-4\cdot\frac{2(2+1)}{2}=27,$

and $h^{1}(S,\mathcal{O}_{S})=0$ since $S$ is rational. Thus,

$h^{0}(C,N_{C|S})=26\mathrm{~{}and~{}}h^{1}(C,N_{C|S})=0$

by Riemann-Roch.

(ii) Claim:

$h^{0}(C,N_{{S|\mathbb{P}^{4}}_{|C}})=h^{0}(C,N_{S|\mathbb{P}^{4}}\otimes\mathcal{O}_{C})=26.$

Since $S$ is a complete intersection of two quadrics, we have

$N_{S|\mathbb{P}^{4}}=\mathcal{O}_{S}(2)\oplus\mathcal{O}_{S}(2),$

and hence

$N_{S|\mathbb{P}^{4}}\otimes\mathcal{O}_{C}=(\mathcal{O}_{S}(2)\oplus\mathcal{O}_{S}(2))\otimes\mathcal{O}_{C}=\mathcal{O}_{C}(2)\oplus\mathcal{O}_{C}(2).$

By Riemann-Roch,

$h^{0}(C,\mathcal{O}_{C}(2))=26-14+1+h^{0}(K_{C}\otimes\mathcal{O}_{C}(-2)).$

Note that

$K_{S}\otimes\mathcal{O}_{S}(C)\otimes\mathcal{O}_{S}(-2)=|(8;3,2^{4})-3(3;1^{5})|=|(-1;0,-1^{4})|$

and by adjunction

$(K_{S}\otimes\mathcal{O}_{S}(C)\otimes\mathcal{O}_{S}(-2))\otimes\mathcal{O}_{C}=K_{C}\otimes\mathcal{O}_{C}(-2),$

we have $h^{0}(K_{C}\otimes\mathcal{O}_{C}(-2))=0$. Hence
$h^{0}(C,\mathcal{O}_{C}(2))=13$ and the claim follows.

(iii) By Claims (i) and (ii) we have

$h^{0}(C,N_{C|\mathbb{P}^{4}})=h^{0}(C,N_{S|\mathbb{P}^{4}}\otimes\mathcal{O}_{C})+h^{0}(C,N_{C|S})=52,$

and hence

$\dim\cal{F}\leq h^{0}(C,N_{C|\mathbb{P}^{4}})=52.$

∎

######  Remark 5.5.

  * (i)

We were not able to determine the irreducibility of $\cal{H}_{13,14,4}$. We
only showed that there is a generically reduced component of the expected
dimension $\cal{X}(13,14,4)=52$. If one can find an irreducible family whose
dimension exceeds $52$, this will lead us to the reducibility of
$\cal{H}_{13,14,4}$.

  * (ii)

One can check that there is no component of $\cal{H}_{13,14,4}$ whose smooth
element lies on a cubic scroll or a normal cubic cone, utilizing standard
computation used in the proof Theorem 5.1, e.g. (2), (3), (4), (5).

  * (iii)

On the other hand, there is only one component of $\cal{H}_{13,14,4}$ whose
general element lies on a quartic surface, which is the one we described in
Proposition 5.4. By [6, Theorem 4.1], curves on a singular del Pezzo is a
specialization of curves on a smooth del Pezzo surface. By a formula similar
to (4), (5) for curves on a cone over an elliptic curve, there is no smooth
curve on an elliptic cone.

  * (iv)

The author tried without success to determine the gonality of the curve
$C\in(11;5,3^{7})$ on
$\mathbb{P}^{2}_{8}\stackrel{{\scriptstyle(4;2,1^{7})}}{{\longrightarrow}}S\subset\mathbb{P}^{4}$,
a smooth curve in $\mathbb{P}^{4}$ with $(d,g)=(13,14)$ whereas $\deg
S=(4;2,1^{7})^{2}=5$. If $C$ is $6$-gonal, this would imply that there is
another component containing $C$, by semi-continuity of gonality. By
Castelnuovo-Severi inequality, a general curve $C\in\cal{F}$ in Proposition
5.4 is $5$-gonal.

  * (v)

Through a tedious computation, it can also be shown that there is no component
of $\cal{H}_{13,14,4}$ whose smooth element lies on a Bordiga surface in
$\mathbb{P}^{4}$.

We finish this section with the following theorem concerning the
irreducibility of $\cal{H}_{2r+5,r+10,r}$ for $r=3$.

######  Theorem 5.6.

The Hilbert scheme $\mathcal{H}_{11.13,3}$ is irreducible of the expected
dimension $44$.

###### Proof.

A smooth $C\subset\mathbb{P}^{3}$ with $(d,g)=(11,13)$ does not lie on a
quadric surface because there is no integer solution for the equations;

$a+b=11=d,~{}~{}(a-1)(b-1)=13=g.$

We also see that $C$ lies on at most one cubic surface, by Bezout. Note that
smooth space curves of degree $d$ and genus $g$ on a smooth cubic surface form
a finite union of locally closed irreducible family in $\mathcal{H}_{d,g,3}$
of dimension $d+g+18$ if $d\geq 10$ by [15, Proposition B.1]. Since
$d+g+18<4d$ for $(d,g)=(11,13)$, the family of curves lying on smooth cubics
does not constitute a component, even though $\cal{H}_{11,13,3}\neq\emptyset$
as we have seen in Theorem 3.2. On the other hand, it is possible that there
might exist a component of $\mathcal{H}_{11,13,3}$ whose general element lies
only on a singular cubic surface. However, one may argue that no such
component exists as follows. Note that every singular cubic surface
$S\subset\mathbb{P}^{3}$ is one of the following three types. (i) $S$ is a
normal cubic surface with some double points only.

(ii) $S$ is a normal cubic cone.

(iii) $S$ is not normal, which may possibly be a cone.

For the case (i), let $S$ be a normal cubic surface which is not a cone. By a
result due to Brevik [9, Theorem 5.24], every curve on $S$ is a specialization
of curves on a smooth cubic surface. Therefore we are done for the case (i).

For the case (ii), we let $C$ be a smooth curve of degree $d$ and genus $g$ on
a normal cubic cone $S$. Recall that

(a) $g=1+d(d-3)/6-2/3$ if $C$ passes through the vertex of $S$

(b) $g=1+d(d-3)/6$, otherwise

which can be found in [15, Proposition 2.12] as an application of C. Segre
formula. However $(d,g)=(11,13)$ satisfies neither of the above. Alternatively
one may check directly that there is no smooth curve of degree $d=11$ and
genus $g=13$ on a cone $S$ over a smooth plane cubic $E\subset\mathbb{P}^{2}$,
realizing $S$ as the image of the ruled surface
$\mathbb{P}(\cal{O}_{E}\oplus\cal{O}_{E}(3))$. (iii) Let $C$ be a smooth curve
of degree $d$ and genus $g$ on a non-normal cubic surface $S$. Recall that if
$S$ is a cone, then $S$ is a cone over a singular plane cubic, in which case
$S$ is a projection of a cone $S^{\prime}$ over a twisted cubic in a
hyperplane in $\mathbb{P}^{4}$ from a point outside $S^{\prime}$. Furthermore,
the minimal desingularisation $\tilde{S}$ of $S^{\prime}$ is isomorphic to the
ruled surface

$\mathbb{F}_{3}=\mathbb{P}(\mathcal{O}_{\mathbb{P}^{1}}\oplus\mathcal{O}_{\mathbb{P}^{1}}(3)).$

If $S$ is not a cone, $S$ is a projection of a rational normal scroll

$S^{\prime\prime}\cong\tilde{S}\cong\mathbb{F}_{1}=\mathbb{P}(\mathcal{O}_{\mathbb{P}^{1}}(1)\oplus\mathcal{O}_{\mathbb{P}^{1}}(2))\subset\mathbb{P}^{4}$

from a point outside $S^{\prime\prime}$. In both cases, we have
$\text{Pic}~{}\tilde{S}=\mathbb{Z}h\oplus\mathbb{Z}f\cong\mathbb{Z}^{\oplus
2}$, where $f$ is the class of a fiber of $\tilde{S}\rightarrow\mathbb{P}^{1}$
and $h=\pi^{*}(\mathcal{O}_{S}(1))$ with
$\tilde{S}\stackrel{{\scriptstyle\pi}}{{\rightarrow}}S$. Note that
$h^{2}=3,f^{2}=0$, $h\cdot f=1$ and $K_{\tilde{S}}\equiv-2h+f$. Denoting by
$\tilde{C}\subset\tilde{S}$ the strict transformation of the curve $C\subset
S$, we set $k:=(\tilde{C}\cdot f)_{\tilde{S}}$. We have $\tilde{C}\equiv
kh+(d-3k)f=kh+(11-3k)f$. By adjunction formula, it follows that

$g=13=\frac{(2\cdot 11-3k-2)(k-1)}{2},$

which does have an integer solution and thus we are done with the case (iii).

Since $\cal{H}_{11,13,3}\neq\emptyset$, we take a component
$\cal{H}\subset\cal{H}_{11,13,3}$. Let $C$ be a general element of $\cal{H}$.
We recall that there is a smooth curve of degree $d=11$ and genus $g=13$ lying
on a smooth irreducible quartic surface by a result of Mori [29, Theorem 1].
We may assume that $C$ does not lie on a quadric or a cubic.

Since every smooth curve $C$ of genus $g=13$ of degree $d=11$ in
$\mathbb{P}^{3}$ lies on at least

$h^{0}(\mathbb{P}^{3},\mathcal{O}(4))-h^{0}(C,\mathcal{O}(4))=35-(4\cdot
11-g+1)=3$ (17)

independent quartics, we see that $C$ is residual to a curve
$D\subset\mathbb{P}^{3}$ of degree $e=5$ and genus $h=1$ in the complete
intersection of two (irreducible) quartics by the basic relation

$2(g-h)=(s+t-4)(d-e).$

Consider the locus

$\Sigma\subset\mathbb{G}(1,\mathbb{P}(H^{0}(\mathbb{P}^{3},\mathcal{O}(4))))=\mathbb{G}(1,34)$

of pencils of quartic surfaces whose base locus consists of a curve $C$ of
degree $d=11$ and genus $g=13$ and an elliptic quintic $D$ which are directly
linked via complete intersection of quartics in the pencil, together with the
two obvious maps

$\begin{array}[]{ccc}\hskip-48.0pt\mathbb{G}(1,34)\supset\Sigma&\stackrel{{\scriptstyle\pi_{C}}}{{\dashrightarrow}}~{}~{}\mathcal{I}_{4}\subset\mathcal{H}_{11,13,3}\\\
\\\
\mathrel{\rotatebox[origin={t}]{-270.0}{\reflectbox{$\dashrightarrow$}}}\vbox{\hbox
to0.0pt{$\scriptstyle{{\pi_{D}}}\,$\hss}}\\\ \\\
\mathcal{H}_{5,1,3}\end{array}$

where $\mathcal{I}_{4}$ is the image of $\Sigma$ under $\pi_{C}$. An elliptic
quintic $D\subset\mathbb{P}^{3}$ lies on at least

$h^{0}(\mathbb{P}^{3},\mathcal{O}(4))-h^{0}(D,\mathcal{O}(4))=35-(4\cdot
5-1+1)=15$ (18)

independent quartics. Note that $D\in\mathcal{H}_{5,1,3}$ is directly linked
to $F\in\mathcal{H}_{4,0,3}$ via complete intersection of two cubics. Since
$F$ is rational, $h^{0}(\mathbb{P}^{3},\mathcal{I}_{F}(2))=1$; otherwise $F$
is elliptic. From
$0\rightarrow\mathcal{I}_{F}(2)\rightarrow\mathcal{O}_{\mathbb{P}^{3}}(2)\rightarrow\mathcal{O}_{F}(2)\rightarrow
0$, we have $h^{1}(\mathbb{P}^{3},\mathcal{I}_{F}(2))=0$. From the well-known
relation

$h^{1}(\mathbb{P}^{3},\mathcal{I}_{C_{1}}(m))=h^{1}(\mathbb{P}^{3},\mathcal{I}_{C_{2}}(s+t-4-m))$

when $C_{1}$ and $C_{2}$ are directly linked by complete intersection of two
surfaces of degrees $s$ and $t$, we have

$h^{1}(\mathbb{P}^{3},\mathcal{I}_{C}(4))=h^{1}(\mathbb{P}^{3},\mathcal{I}_{D})=h^{1}(\mathbb{P}^{3},\mathcal{I}_{F}(2))=0$
(19)
$h^{1}(\mathbb{P}^{3},\mathcal{I}_{D}(4))=h^{1}(\mathbb{P}^{3},\mathcal{I}_{F}(-2))=0.$
(20)

By (20) and (18), we have $h^{0}(\mathbb{P}^{3},\cal{I}_{D}(4))=15$ and
therefore $\pi_{D}$ is generically surjective with fibers open subsets of
$\mathbb{G}(1,14)$. Since $\dim\mathcal{H}_{5.1.3}$ is known to be irreducible
(cf.[14] or [16, page 54]), $\Sigma$ is irreducible and

$\dim\Sigma=\dim\mathbb{G}(1,14)+\dim\mathcal{H}_{5,1,3}=26+4\cdot 5=46.$

On the other hand, by (19) and (17) every smooth curve $C\in\mathcal{I}_{4}$
lies on exactly $3$ independent quartics and hence $\pi_{C}$ is generically
surjective with fibers open subsets of $\mathbb{G}(1,2)$. Finally it follows
that $\mathcal{H}_{11.13.3}$ is irreducible of dimension

$\dim\Sigma-\dim{\mathbb{G}}(1,2)=4\cdot 11.$

∎

##  6\. Irreducibility of $\mathcal{H}^{\mathcal{L}}_{2r+6,r+11,r}$

In this section we determine the irreducibility of
$\mathcal{H}^{\mathcal{L}}_{2r+6,r+11,r}$ for $7\leq r\leq 11$. Recall that
$\mathcal{H}^{\mathcal{L}}_{2r+6,r+11,r}\neq\emptyset$ only if $3\leq r\leq
11$ by Theorem 3.2 (c). The following result employs a similar method we used
in Theorem 5.1. However, there appear some subtleties which need some caution.

######  Theorem 6.1.

$\mathcal{H}^{\mathcal{L}}_{2r+6,r+11,r}$ is irreducible if $r=8,9,11$ and
reducible if $r=10$.

###### Proof.

Given a component
$\mathcal{G}\subset{\widetilde{\mathcal{G}}}_{\mathcal{L}}\subset\mathcal{G}^{r}_{2r+6}$,
choose a general $\mathcal{D}\in\mathcal{G}$ and set
$\mathcal{E}=\mathcal{D}^{\vee}=g^{4}_{14}$. We claim that $\mathcal{E}$ is
base-point-free and birationally very ample. If $\mathcal{E}$ is compounded,
we apply Lemma 2.10 to deduce that $\mathcal{E}^{\vee}=\mathcal{D}$ is not
very ample. $\mathcal{E}$ is base-point-free since $\pi(13,4)=18<g=r+11$.

In case $8\leq r\leq 11$,

$\pi_{1}(14,4)=18<g=r+11\leq
p_{a}(C_{\mathcal{E}})\leq\pi(14,4)=22~{}~{}\hskip 12.0pt$

hence $C_{\mathcal{E}}\subset\mathbb{P}^{4}$ is an extremal or a nearly
extremal curve lying on a cubic surface $S\subset\mathbb{P}^{4}$.

(A) We first assume $S\cong\mathbb{F}_{1}\cong\mathbb{P}^{2}_{1}$. Set

$\delta=\delta(p_{a}(C_{\mathcal{E}})):=p_{a}(C_{\mathcal{E}})-g=p_{a}(C_{\mathcal{E}})-(r+11)\geq
0$

and we list up all the pairs $(a,b)\in\mathbb{N}\times\mathbb{Z}$ satisfying
the equations (2), (3) in Remark 2.3; $n=3,d=14$ with $19\leq
p_{a}(C_{\mathcal{E}})\leq 22$;

$\begin{cases}p_{a}(C_{\mathcal{E}})=22;~{}(a,b)=(5,-1),&8\leq r\leq 11\\\
p_{a}(C_{\mathcal{E}})=21;~{}(a,b)=(4,2),&8\leq r\leq 10\\\
p_{a}(C_{\mathcal{E}})=20;~{}(a,b)=(6,-4),&8\leq r\leq 9\\\
p_{a}(C_{\mathcal{E}})=19;~{}\mathrm{no~{}solution}\\\
p_{a}(C_{\mathcal{E}})=18;~{}\mathrm{no~{}solution}\end{cases}$ (21)

Denoting the three linear systems on $S$ by

$\mathcal{M}_{1}=|5H-L|,\mathcal{M}_{2}=|4H+2L|,\mathcal{M}_{3}=|6H-4L|,$

we have the following possibilities;

$\begin{cases}r=11;&C_{\mathcal{E}}\in\mathcal{M}_{1},~{}\delta=0\\\
r=10;&C_{\mathcal{E}}\in\mathcal{M}_{1},~{}\delta=1~{}\textrm{ or
}~{}C_{\mathcal{E}}\in\mathcal{M}_{2},~{}\delta=0\\\
r=~{}9;&C_{\mathcal{E}}\in\mathcal{M}_{1},~{}\delta=2~{}\textrm{ or
}~{}C_{\mathcal{E}}\in\mathcal{M}_{2},~{}\delta=1~{}\textrm{ or
}~{}C_{\mathcal{E}}\in\mathcal{M}_{3},~{}\delta=0\\\
r=~{}8;&C_{\mathcal{E}}\in\mathcal{M}_{1},~{}\delta=3~{}\textrm{ or
}~{}C_{\mathcal{E}}\in\mathcal{M}_{2},~{}\delta=2~{}\textrm{~{} or
~{}}~{}C_{\mathcal{E}}\in\mathcal{M}_{3},~{}\delta=1.\end{cases}$

We need to check, given a fixed $r$ and for each possible value
$p_{a}(C_{\mathcal{E}})$, what linear systems $\mathcal{M}_{i}$ such that
$C_{\mathcal{E}}\in\mathcal{M}_{i}$ contribute to a component
$\mathcal{G}\subset\widetilde{\mathcal{G}}_{\mathcal{L}}\subset\mathcal{G}^{r}_{2r+6}$
whose general element is complete and very ample. In other words, we are
asking; for what $\mathcal{M}_{i}$ with $C_{\mathcal{E}}\in\mathcal{M}_{i}$,
is the linear system

$|D|:=|K_{\mathbb{P}^{2}_{\delta+1}}+\widetilde{C}_{\cal{E}}-\widetilde{H}|_{{|\widetilde{C}_{\cal{E}}}}$

very ample ?

Here $\widetilde{C}_{\cal{E}}$ is the proper transformation of
$C_{\mathcal{E}}\in\Sigma_{{\mathcal{{M}}_{i}},\delta}$ under the blow up of
$\mathbb{F}_{1}\cong\mathbb{P}^{2}_{1}\cong S$ at
$\delta=p_{a}(C_{\mathcal{E}})-g=p_{a}(C_{\mathcal{E}})-(r+11)$ nodal points
of $C_{\mathcal{E}}$,
$\mathbb{P}^{2}_{\delta+1}\stackrel{{\scriptstyle\pi}}{{\rightarrow}}\mathbb{F}_{1}\cong
S$ and $\widetilde{H}=\pi^{*}\mathcal{O}_{S}(1)$.

  * (i)

$p_{a}({C_{\mathcal{E}}})=22\Longleftrightarrow
C_{\mathcal{E}}\in\mathcal{M}_{1}$; every $8\leq r\leq 11$ is possible and
$\delta=11-r$. Since

$C_{\mathcal{E}}\in\mathcal{M}_{1}=|5H-L|=|5(e+2f)-f|=|9l-4e|=(9;4),$

we have

$\widetilde{C}_{\cal{E}}\in(9,4,2^{11-r})$

on $\mathbb{P}^{2}_{\delta+1}=\mathbb{P}^{2}_{12-r}$ – which is a special case
of the linear system (10) in Theorem 3.2 – and

$\displaystyle|K_{\mathbb{P}^{2}_{12-r}}+\widetilde{C}_{\cal{E}}-\widetilde{H}|$
$\displaystyle=|(-3;-1^{12-r})+(9;4,2^{11-r})-(2,1,0^{12-r})|$
$\displaystyle=|(4;2,1^{11-r})|,$

which is very ample666$|(4;2,1^{11-r})|$ is very ample not only for $8\leq
r\leq 11$ but also for $4\leq r\leq 11$.. Hence we have an irreducible family
of nodal curves lying on a Hirzebruch surface $\mathbb{F}_{1}$ – parametrized
by the irreducible Severi variety $\Sigma_{\mathcal{M}_{1},11-r}$ – whose non-
singular model

$\widetilde{C}_{\cal{E}}\subset\mathbb{P}^{2}_{12-r}\lhook\joinrel\xrightarrow{(4;2,1^{11-r})}\mathbb{P}^{r}$

is embedded into $\mathbb{P}^{r}$ by the linear system
$|K_{\mathbb{P}^{2}_{12-r}}+\widetilde{C}_{\cal{E}}-\widetilde{H}|_{|\widetilde{C}_{\cal{E}}}$
as a linearly normal curve of degree

$d=(K_{\mathbb{P}^{2}_{12-r}}+\widetilde{C}_{\cal{E}}-\widetilde{H})\cdot\widetilde{C}_{\cal{E}}=2r+6.$

Therefore we may conclude that for every $8\leq r\leq 11$ and
$\delta=p_{a}(C_{\mathcal{E}})-g=11-r$, there is a component
$\mathcal{G}_{\cal{M}_{1},11-r}\subset\mathcal{G}^{r}_{2r+6}$ associated with
the Severi variety $\Sigma_{\mathcal{M}_{1},11-r}$, i.e

$\mathcal{G}_{\cal{M}_{1},11-r}:={\widetilde{\Sigma}}_{\mathcal{M}_{1},11-r}^{\vee}={\Sigma}_{\mathcal{M}_{1},11-r}^{\vee}/\operatorname{Aut}(S),~{}\mathrm{where}$
$\Sigma_{\mathcal{M}_{1},11-r}^{\vee}:=\\{~{}|K_{\mathbb{P}^{2}_{12-r}}+\widetilde{C}_{\cal{E}}-\widetilde{H}|_{|{\widetilde{C}_{\cal{E}}}}~{}|~{}C_{\mathcal{E}}\in\Sigma_{\mathcal{M}_{1},\delta}\\}.$

Note that a general $C_{\mathcal{E}}\in\Sigma_{\mathcal{M}_{1},11-r}$ is
$5$-gonal by Castelnuovo-Severi inequality.

  * (ii)

$p_{a}({C_{\mathcal{E}}})=21\Longleftrightarrow
C_{\mathcal{E}}\in\mathcal{M}_{2}$; only $8\leq r\leq 10$ is possible and
$\delta=10-r$. On $S\cong\mathbb{P}^{2}_{1}$, we have

$C_{\mathcal{E}}\in\mathcal{M}_{2}=|4H+2L|=|4(e+2f)+2f|=|10l-6e|=(10;6).$

Thus we have

$\widetilde{C}_{\cal{E}}\in(10;6,2^{10-r})$

after blowing up $\mathbb{P}^{2}_{1}$ at $\delta=10-r$ nodal points of
$C_{\mathcal{E}}$ and

$\displaystyle|K_{\mathbb{P}^{2}_{11-r}}+\widetilde{C}_{\cal{E}}-\widetilde{H}|$
$\displaystyle=|(-3;-1^{11-r})+(10;6,2^{10-r})-(2,1,0^{10-r})|$
$\displaystyle=|(5;4,1^{10-r})|.$

Note that both $|\widetilde{C}_{\cal{E}}|=(10;6,2^{10-r})$ and
$|K_{\mathbb{P}^{2}_{11-r}}+\widetilde{C}_{\cal{E}}-\widetilde{H}|=(5;4,1^{10-r})$
are very ample only if $r=10$ by [13]. Hence only for the case $r=10$, there
is an irreducible family
$\mathcal{G}_{\cal{M}_{2},0}\subset\widetilde{\mathcal{G}}_{\mathcal{L}}\subset\mathcal{G}^{r}_{2r+6}$
associated with the Severi variety
$\Sigma_{{\mathcal{M}_{2}},10-r}=\Sigma_{\mathcal{M}_{2},0}$, i.e.

$\mathcal{G}_{\cal{M}_{2},0}={\widetilde{\Sigma}}_{\mathcal{M}_{2},0}^{\vee}.$

Therefore we obtain another irreducible family of (smooth) curves on
$\mathbb{F}_{1}$ which is embedded into $\mathbb{P}^{10}$ by
$|K_{\mathbb{P}^{2}_{1}}+{C_{\mathcal{E}}}-\mathcal{O}_{S}(1)|=(5;4)$ as a
linearly normal curve with $(d,g)=(2r+6,r+11)=(26,21)$. By Castelnuovo-Severi
inequality, a general
$C_{\mathcal{E}}\in\Sigma_{{\mathcal{M}_{2}},10-r}=\Sigma_{\mathcal{M}_{2},0}$
is $4$-gonal. For $r=10$, we have

$\dim\mathcal{G}_{\cal{M}_{1},11-r}=\dim\mathcal{M}_{1}-1-\dim\textrm{Aut}(S)=37<\dim\mathcal{G}_{\cal{M}_{2},0}=38,$

thus $\mathcal{G}_{\cal{M}_{1},1}$ and $\mathcal{G}_{\cal{M}_{2},0}$ are two
distinct components of $\widetilde{\mathcal{G}}_{\mathcal{L}}$ by semi-
continuity of gonality.

  * (iii)

$p_{a}({C_{\mathcal{E}}})=20\Longleftrightarrow
C_{\mathcal{E}}\in\mathcal{M}_{3}=|6H-4L|$; only $8\leq r\leq 9$ is possible.
By the same computation as before, after blowing up $\mathbb{P}^{2}_{1}$ at
$\delta(p_{a}(C_{\mathcal{E}}))=p_{a}(C_{\mathcal{E}})-g=9-r$ nodal points of
$C_{\mathcal{E}}$, we get on $\mathbb{P}^{2}_{10-r}$

$\widetilde{C}_{\cal{E}}\in(8;2,2^{9-r})=(8;2^{10-r})$

and

$\displaystyle|K_{\mathbb{P}^{2}_{10-r}}+\widetilde{C}_{\cal{E}}-\widetilde{H}|$
$\displaystyle=(-3;-1^{10-r})+(8;2^{10-r})-(2,1,0^{9-r})$
$\displaystyle=(3;0,1^{9-r}),$

which is not very ample. Hence
$|D|=|K_{\mathbb{P}^{2}_{\delta+1}}+\widetilde{C}_{\cal{E}}-\widetilde{H}|_{\widetilde{|C_{\mathcal{E}}}}$
is not very ample.777For every $i=1,2,3$ with
${C_{\mathcal{E}}}\in\mathcal{M}_{i}$, the restriction map
$\rho:H^{0}(\mathbb{P}^{2}_{\delta+1},\mathcal{O}(K_{\mathbb{P}^{2}_{\delta+1}}+\widetilde{C}_{\cal{E}}-\widetilde{H}))\rightarrow
H^{0}(\widetilde{C}_{\cal{E}},\mathcal{O}(K_{\mathbb{P}^{2}_{\delta+1}}+\widetilde{C}_{\cal{E}}-\widetilde{H})\otimes\cal{O}_{\widetilde{C}_{\cal{E}}})$
is surjective since
$h^{1}(\mathbb{P}^{2}_{\delta+1},\mathcal{O}(K_{\mathbb{P}^{2}_{\delta+1}}-\widetilde{H}))=0$.

Summing up, all the possible components of
$\widetilde{\mathcal{G}}_{\mathcal{L}}\subset\mathcal{G}^{r}_{2r+6}$ such that
$C_{\mathcal{E}}$ lies on a smooth cubic scroll in $\mathbb{P}^{4}$ are the
following.

$\begin{cases}r=11;&\mathcal{G}_{\cal{M}_{1},0}\\\
r=10;&\mathcal{G}_{\cal{M}_{1},1},~{}\mathcal{G}_{\cal{M}_{2},0}\\\
r=~{}9;&\mathcal{G}_{\cal{M}_{1},2}\\\
r=~{}8;&\mathcal{G}_{\cal{M}_{1},3}\end{cases}$

(B) We assume that $C_{\mathcal{E}}$ lies on a rational normal cone
$S\subset\mathbb{P}^{4}$. We borrow the notation from Remark 2.3 (iii) and
take $n=3$, $d=14$ there. Let $\widetilde{C}_{\cal{E}}$ be the proper
transformation of $C_{\mathcal{E}}$ under the minimal desingularization
$\mathbb{F}_{3}\xrightarrow{\ \ }S\subset\mathbb{P}^{4}$ given by
$|C_{0}+3f|$. Again by (5) and (4) in Remark 2.3, we have

$(k,m,p_{a}(\widetilde{C}_{\cal{E}}))=\begin{cases}(4,2,21)\\\ (3,5,17)\\\
(2,8,10).\end{cases}$

For the second and the third triples, $p_{a}(\widetilde{C}_{\cal{E}})<g=r+11$
for any $8\leq r\leq 11$, hence only the first triple
$(k,m,p_{a}(\widetilde{C}_{\cal{E}}))=(4,2,21)$ is possible. (B-1): If $r=11$,
$g(\widetilde{C}_{\cal{E}})=g=r+11=22>p_{a}(\widetilde{C}_{\cal{E}})$ a
contradiction. Thus there is no curve (singular or smooth) $C_{\mathcal{E}}$
with geometric genus $g=r+11=22$ and degree $14$ on the rational normal cone
$S\subset\mathbb{P}^{4}$.

(B-2): If $r=10$,

$p_{a}(\widetilde{C}_{\cal{E}})=21=g=r+11=g(C_{\mathcal{E}})=g(\widetilde{C}_{\cal{E}}),$

hence $\widetilde{C}_{\cal{E}}$ is smooth. On the other hand, the image
$C_{\mathcal{E}}$ of $\widetilde{C}_{\cal{E}}$ under the morphism
$\mathbb{F}_{3}\xrightarrow{\ \ }S\subset\mathbb{P}^{4}$ is singular since
$m=2$ and hence $g=g(C_{\mathcal{E}})<g(\widetilde{C}_{\cal{E}})=21$, which is
a contradiction. Again, there is no curve (singular or smooth)
$C_{\mathcal{E}}$ with geometric genus $g=r+11=22$ and degree $14$ on the
rational normal cone $S\subset\mathbb{P}^{4}$.

(B-3): If $r=9$, $\widetilde{C}_{\cal{E}}$ is singular with one double point;
$\widetilde{C}_{\cal{E}}\in|4C_{0}+14f|$.

(B-4): If $r=8$, $\widetilde{C}_{\cal{E}}$ is singular with a double point and
an additional singular (possibly infinitely near) point;
$\widetilde{C}_{\cal{E}}\in|4C_{0}+14f|$.

Note that we are under the condition
$(k,m,p_{a}(\widetilde{C}_{\cal{E}}))=(4,2,21)$, thus ${C}{E}$ is singular at
the vertex of the cone $S\subset\mathbb{P}^{4}$. Hence at least one singular
point of $\widetilde{C}{E}$ lies on the section $C_{0}$. However, in order to
avoid complicated computation and unnecessary extra notation, we will show
that a general element in the Severi variety of curves of geometric genus
$g=r+11\,(8\leq r\leq 9)$ in the linear system $\mathcal{N}=|4h+14f|$ on
$\mathbb{F}_{3}$ does not have very ample $|K_{\widehat{C}}(-1)|$ where
$\widehat{C}$ is the normalization of
$\widetilde{C}_{\cal{E}}\subset\mathbb{F}_{3}$. This simplification can be
justified because being very ample is an open condition.

We did a similar job in the proof (part (B-2)) of Theorem 5.1 for one node
case. Even though all the arguments are parallel to part (B-2) in Theorem 5.1,
we provide an outline of a proof for the two nodes case for the convenience of
readers. We assume that $\widetilde{C}_{\cal{E}}\subset\mathbb{F}_{3}$ has two
nodes outside $C_{0}$ as its only singularities. Let $e_{i}(\,i=1,2)$ be
exceptional divisors and let $f_{i}\,(i=1,2)$ be the fibers containing the two
nodal points of $\widetilde{C}_{\cal{E}}$. After resolving the two nodes we
get a smooth curve ${\widehat{C}}\subset\mathbb{F}_{3,2}$ on the surface
$\mathbb{F}_{3}$ blown up at two points. We have

${\widehat{C}}\in|4C_{0}+14f-\sum 2e_{i}|.$

Set

$\displaystyle\mathcal{M}:$
$\displaystyle=|{\widehat{C}}+K_{{\mathbb{F}_{3,2}}}-(C_{0}+3f)|$
$\displaystyle=|(4C_{0}+14f-\sum 2e_{i})+(-2C_{0}-5f+\sum e_{i})-(C_{0}+3f)|$
$\displaystyle=|C_{0}+6f-\sum e_{i}|.$

The injectivity of the restriction map

$\rho:H^{0}({{\mathbb{F}_{3,2}}},\mathcal{O}(\mathcal{M}))\longrightarrow
H^{0}({\widehat{C}},\mathcal{O}(\mathcal{M})\otimes\mathcal{O}_{{\widehat{C}}})$

follows from

$h^{0}(\mathbb{F}_{3,2},\mathcal{O}_{{\mathbb{F}_{3,2}}}(\mathcal{M}-{\widehat{C}}))=h^{0}(\mathbb{F}_{3,2},\mathcal{O}_{{\mathbb{F}_{3,2}}}(-(3C_{0}+8f-\sum
e_{i})))=0.$

The surjectivity of the restriction map $\rho$ follows from Castelnuovo genus
bound for a curve of degree $\widehat{C}\cdot\mathcal{M}=(4C_{0}+14f-\sum
2e_{i})\cdot(C_{0}+6f-\sum e_{i})=22$ in $\mathbb{P}^{r\geq 9}$ unless $\rho$
is surjective; $\pi(22,9)=18<g=19$. Therefore we have

$\mathrm{Im}(\rho)=H^{0}(\widehat{C},\mathcal{O}(\mathcal{M})\otimes\mathcal{O}_{\widehat{C}})$

and the restriction map $\rho$ is an isomorphism. Denoting by
$\widetilde{f}_{i}$ the proper transformation of $f_{i}$ under the blow up,

$\tilde{f}_{i}^{2}=-1,\tilde{f}_{i}\cdot
e_{i}=1,C_{0}\cdot\tilde{f}_{i}=1,{\widehat{C}}\cdot\tilde{f}_{i}=2,\cal{M}\cdot\tilde{f}_{i}=(C_{0}+6f-\Sigma
e_{i})\cdot\tilde{f}_{i}=0.$

Hence the morphism $\psi$ given by
$\mathcal{M}=|{\widehat{C}}+K_{{\mathbb{F}_{3,2}}}-(C_{0}+3f)|$ contracts
$(-1)$ curves $\tilde{f}_{i}$ and the image curve
$\psi({\widehat{C}})\subset\mathbb{P}^{5}$ acquires singularities. Since the
complete linear system $\mathrm{Im}(\rho)$ maps ${\widehat{C}}$ onto
$\psi({\widehat{C}})\subset\mathbb{P}^{5}$ with at least $2$ singular points,
$\mathcal{M}\otimes\mathcal{O}_{{\widehat{C}}}=|K_{{\widehat{C}}}(-1)|$ is not
very ample. Conclution: Part (B) shows that there is no component of
$\mathcal{H}^{\mathcal{L}}_{2r+6,r+11,r}$ whose general element lies on a
rational normal cone and this finishes the proof. ∎

We would like to push forward one step further to determine the reducibility
of $\mathcal{H}^{\mathcal{L}}_{2r+6,r+11,r}$ for $r=7$. In this case, the
curve $C_{\mathcal{E}}\subset\mathbb{P}^{4}$ may not be nearly extremal;
$\pi_{1}(14,4)=18=r+11=g$. Therefore we cannot use a similar method which we
used in Theorem 6.1. We instead estimate the dimension of a component
containing a certain (natural) irreducible family of curves by computing the
normal bundle corresponding to a general element of the family. This is the
point where a certain subtlety arises. One drawback of this method is that we
cannot identify all the irreducible components as completely as we did before,
unless we carry out lengthy and cumbersome analysis of all the possibilities
with our imperfect techniques.

######  Proposition 6.2.

$\mathcal{H}^{\mathcal{L}}_{2r+6,r+11,r},r=7$ is reducible.

###### Proof.

We retain almost all the notations which were used throughout this section.
Choose a general
$\mathcal{D}\in\mathcal{G}\subset\widetilde{\mathcal{G}}\subset\mathcal{G}^{r}_{2r+6}$
and set $\mathcal{E}=\mathcal{D}^{\vee}=g^{4}_{14}$. Claim: $\mathcal{E}$ is
birationally very ample with possibly non-empty base locus $\Delta$ such that
$0\leq\deg\Delta\leq 1$. If $\mathcal{E}$ is compounded inducing
$C\stackrel{{\scriptstyle\eta}}{{\rightarrow}}E$, we apply Clifford’s theorem
to the complete linear system $\widetilde{\mathcal{E}}$ on the base curve $E$
such that $\eta^{*}{\widetilde{\mathcal{E}}}=|\mathcal{E}-\Delta|$. By Lemma
2.10, $\cal{D}=\mathcal{E}^{\vee}$ is not very ample, a contradiction. If
$\deg\Delta\geq 2$, we have $\pi(12,4)=15<g=r+11=18$. Hence $\deg\Delta\leq 1$
and $\mathcal{E}$ is birationally very ample.

(A) $\deg\Delta=1$: Set
$\mathcal{E}^{\prime}:=|\mathcal{E}-\Delta|=g^{4}_{13}$. Since
$\pi(13,4)=18=g=r+11$, the curve
$C_{\mathcal{E}^{\prime}}\subset\mathbb{P}^{4}$ induced by
$\mathcal{E}^{\prime}$ is an extremal (smooth) curve lying on a cubic surface
$S\subset\mathbb{P}^{4}$. We may assume that $S$ is smooth by Remark 2.5. By
solving (2), (3), we have either $C_{\mathcal{E}^{\prime}}\in|5H-2L|$ or
$C_{\mathcal{E}^{\prime}}\in|4H+L|$. Noting that $C_{\cal{E}^{\prime}}\cong
C$, we have

$|K_{C_{\mathcal{E}^{\prime}}}-\mathcal{E}^{\prime}-\Delta|=|K_{C_{\mathcal{E}^{\prime}}}-\mathcal{E}|=|K_{C}-\mathcal{E}|=\mathcal{D},~{}~{}|K_{C_{\mathcal{E}^{\prime}}}-\mathcal{E}^{\prime}|=g^{8}_{21}.$

Thus our $C\subset\mathbb{P}^{7}$ with $(d,g)=(20,18)$ is the image of the
projection with center at $\Delta$ from the (smooth) image curve of the
morphism induced by $|K_{C_{\mathcal{E}^{\prime}}}-\mathcal{E}^{\prime}|$.

(A-1) $C_{\mathcal{E}^{\prime}}\in|5H-2L|$: We have
$|K_{S}+C_{\mathcal{E}^{\prime}}-H|=|2H-L|=|3l-e|$; cf. (5). Let
$T\subset\mathbb{P}^{8}$ be the image of the embedding
$S\stackrel{{\scriptstyle\varphi}}{{\longrightarrow}}T\subset\mathbb{P}^{8}$
induced by $|2H-L|$ and set $C^{\prime}=\varphi(C_{\cal{E}^{\prime}})\subset
T$. Note that $T$ is a smooth del-Pezzo surface of degree $(2H-L)^{2}=8$.
Since there is only finitely many lines on a smooth del-Pezzo of degree $8$,
projecting $C^{\prime}$ into $\mathbb{P}^{7}$ from a general point $p\in
C^{\prime}\subset\mathbb{P}^{8}$ induces an isomorphism onto a curve
$C\subset\mathbb{P}^{7}$, $\deg C=20$. Set

$\mathcal{F}:=\\{~{}g^{8}_{21}~{}|~{}g^{8}_{21}=|K_{C_{\mathcal{E}^{\prime}}}-\mathcal{E}^{\prime}|,\,\mathcal{E}^{\prime}=\mathcal{E}\setminus\Delta,\,\mathcal{E}=\mathcal{D}^{\vee},\,\mathcal{D}\in\mathcal{G}\\}\subset\mathcal{G}^{8}_{21}.$

By construction we have

$\dim\mathcal{F}=\dim|5H-2L|-\dim\operatorname{Aut}(S)=32.$

Setting
$\mathcal{F}_{0}:=\\{~{}|g^{8}_{21}-p|~{}|~{}g^{8}_{21}\in\mathcal{F},~{}p\in
C^{\prime}=\varphi(C_{\cal{E}^{\prime}})\\}\subset\cal{G}^{7}_{20},$ we have

$\dim\mathcal{F}_{0}=\dim\mathcal{F}+1=33>\lambda(14,18,4)=\lambda(20,18,7)=29.$

(A-2) $C_{\mathcal{E}^{\prime}}\in|4H+L|$: Both $|4H+L|$ and
$|K_{S}+C_{\mathcal{E}^{\prime}}-H|=|H+2L|$ are very ample. Under the
isomorphism
$S\stackrel{{\scriptstyle|H+2L|}}{{\longrightarrow}}T\subset\mathbb{P}^{8}$,
we have

$\deg T=(H+2L)^{2}=7$

and hence $T$ is a rational normal surface scroll. Since $(H+2L)\cdot L=1$ and
$(4H+L)\cdot L=4$, the image in $T$ of the ruling $|L|$ on $S$ cut out a
$4$-secant line on the image curve of $C_{\cal{E}^{\prime}}$. Hence the
projection with center at $\Delta$, i.e. the morphism induced by
$\cal{D}=\mathcal{E}^{\vee}=\mathcal{E}^{\prime\vee}-\Delta$ is not very
ample. (B) $\Delta=\emptyset$: Since $\pi_{1}(14,4)=18=r+11=g$,
$C_{\mathcal{E}}\subset\mathbb{P}^{4}$ may lie either on a cubic surface or on
a quartic surface. (B-1) Assume $C_{\mathcal{E}}$ lies on a smooth cubic
scroll $S\subset\mathbb{P}^{4}$. We have

$g=\pi_{1}(14,4)=18\leq p_{a}(C_{\mathcal{E}})\leq 22=\pi(14,4).$

The pairs $(a,b)\in\mathbb{N}\times\mathbb{Z}$ satisfying (2),(3) for $18\leq
p_{a}(C_{\mathcal{E}})\leq 22$ are already listed in (21) (in the proof of
Theorem 6.1).

We take $C_{\mathcal{E}}\in\mathcal{M}_{1}=|5H-L|$, where
$p_{a}(C_{\mathcal{E}})=22$, $\delta=\delta(p_{a}(C_{\mathcal{E}}))=11-r=4$.
The proper transform
$\widetilde{C}_{\cal{E}}\in|(9;4,2^{4})|\in\textrm{Pic}(\mathbb{P}^{2}_{5})$
under the blow up
$\mathbb{P}^{2}_{\delta(p_{a}(C_{\mathcal{E}}))+1}\stackrel{{\scriptstyle\pi}}{{\longrightarrow}}\mathbb{F}_{1}\cong
S$ is embedded into $\mathbb{P}^{7}$ as a curve of degree $d=20$ by the very
ample $(4;2,1^{4})$; cf. part (A)–(i) in the proof of Theorem 6.1.

Therefore the irreducible family $\mathcal{F}_{1}\subset\mathcal{G}^{7}_{20}$
arising this way, more pecisely,

$\cal{F}_{1}={\widetilde{\Sigma}}_{\mathcal{M}_{1},11-r}^{\vee}=\\{~{}|K_{\mathbb{P}^{2}_{12-r}}+\widetilde{C}_{\cal{E}}-\widetilde{H}|_{|{\widetilde{C}_{\cal{E}}}}~{}|~{}C_{\mathcal{E}}\in\Sigma_{\mathcal{M}_{1},\delta}\\}/\operatorname{Aut}(S).$

has dimension;

$\displaystyle\dim\mathcal{F}_{1}$
$\displaystyle=\dim{\widetilde{\Sigma}}_{\mathcal{M}_{1},11-r}^{\vee}=\dim{\Sigma}_{\mathcal{M}_{1},11-r}^{\vee}/\operatorname{Aut}(S)$
$\displaystyle=\dim\Sigma_{\mathcal{M}_{1},11-r}-\dim\textrm{Aut}(S)=\dim|5H-L|-\delta-6=34.$

In the proof of Theorem 6.1, there are several other linear systems
$\cal{M}_{i}$ ($i=2,3$) such that $C_{\mathcal{E}}\in\cal{M}_{i}$. However we
already verified that $C_{\mathcal{E}}^{\vee}\subset\mathbb{P}^{7}$ (the
residual curve of $C_{\mathcal{E}}$) is not smooth.

(B-2) Assume $C_{\mathcal{E}}$ lies on a smooth del Pezzo surface
$S\subset\mathbb{P}^{4}$ and let $C_{\mathcal{E}}\in(a;b_{1},\cdots,b_{5})$.
Solving

$\deg
C_{\mathcal{E}}=3a-\sum^{5}_{i=1}b_{i}=14,~{}C_{\mathcal{E}}^{2}=a^{2}-\sum^{5}_{i=1}b_{i}^{2}=2g-2-K_{S}\cdot
C_{\mathcal{E}}=48$

we get

$(a;b_{1},\cdots,b_{5})=(10;4,3^{4})\textrm{ or }(11,4^{4},3).$ (22)

Note that $(10;4,3^{4})\textrm{ and }(11,4^{4},3)$ are very ample. Also note
that a general member in $(10;4,3^{4})$ is isomorphic to a general member in
$(11,4^{4},3)$ via obvious quadratic transformation.

Let $C_{\mathcal{E}}\in(10;4,3^{4})$. Since

$|{C_{\mathcal{E}}}+K_{S}-H|=(10;4,3^{4})-2(3;1^{5})=(4;2,1^{4})$

is very ample, there is another irreducible family
$\mathcal{F}_{2}\subset\mathcal{G}^{7}_{20}$ whose the residual series
$\cal{D}^{\vee}=\cal{E}$ of a general member $\mathcal{D}\in\mathcal{F}_{2}$
gives rise to a smooth curve $C_{\mathcal{E}}$ on a del Pezzo surface in
$\mathbb{P}^{4}$ and $\mathcal{E}^{\vee}$ induces an embedding into
$\mathbb{P}^{r}$ with $r=\dim|(4;2,1^{4})|=7$, i.e.

$\cal{F}_{2}=\\{\,g^{7}_{20}=\cal{D}\,|\,\cal{E}=\cal{D}^{\vee},\,C_{\mathcal{E}}\subset
S\subset\mathbb{P}^{4},S\mathrm{~{}is~{}a~{}smooth~{}del~{}Pezzo}\\}$

Since a smooth del Pezzo surface in $\mathbb{P}^{4}$ is a complete
intersection of two quadric hypersurfaces, which in turn corresponds to an
element of the Grassmannian
$\mathbb{G}(1,\mathbb{P}(H^{0}(\mathbb{P}^{4},\cal{O}(2)))$, an easy dimension
count yields

$\dim\mathcal{F}_{2}=\dim|(10;4,3^{4})|+\dim\mathbb{G}(1,14)-\dim\operatorname{Aut}(\mathbb{P}^{4})=33.$

So far we have the following three irreducible families which consist very
ample linear series’ inside $\mathcal{G}^{7}_{20}$ having dimensions greater
than $\lambda(20,18,7)=\lambda(14,18,4)=29$.

  1. (0)

$\mathcal{F}_{0}$: $\dim\mathcal{F}_{0}=33$; for $\cal{D}\in\cal{F}_{0}$,
$\mathcal{E}=\cal{D}^{\vee}$ has non-empty base locus,
$C_{\mathcal{E}^{\prime}}\subset\mathbb{P}^{4}$ induced by the moving part
$\mathcal{E}^{\prime}$ of $\mathcal{E}$ is a smooth curve of degree $13$ lying
on a rational normal scroll $S$. $C_{\mathcal{E}^{\prime}}$ is $5$-gonal.
$\mathcal{E}^{\prime\vee}$ is very ample.

  2. (1)

$\mathcal{F}_{1}$: $\dim\mathcal{F}_{1}=34$;
$C_{\mathcal{E}}\subset\mathbb{P}^{4}$ induced by $\mathcal{E}=\cal{D}^{\vee}$
is singular lying on a rational normal scroll $S$ and $C_{\mathcal{E}}$ is
$5$-gonal.

  3. (2)

$\mathcal{F}_{2}$: $\dim\mathcal{F}_{2}=33$;
$C_{\mathcal{E}}\subset\mathbb{P}^{4}$ induced by $\mathcal{E}=\cal{D}^{\vee}$
is a smooth curve lying on a smooth del Pezzo surface in $\mathbb{P}^{4}$ and
$C_{\mathcal{E}}$ is $6$-gonal.888$C_{\mathcal{E}}\in(10;4,3^{4})$ being
$6$-gonal is not totally clear. One may consult [31, Theorem 2] for a quick
check.

Observation:

  * (i)

Note that $\mathcal{F}_{2}$ is not in the boundary of $\mathcal{F}_{1}$; by
lower semi-continuity of gonality or by the fact that singular curves cannot
be specialized to a smooth curve.

  * (ii)

A general $\cal{F}_{2}$ is $6$-gonal. Therefore $\cal{F}_{2}$ is not in the
boundary of $\cal{F}_{0}$.

  * (iii)

Since $\dim\cal{F}_{0}=\dim\cal{F}_{2}$, $\cal{F}_{0}$ is not in the boundary
of $\cal{F}_{2}$.

Suppose that Hilbert scheme
$\mathcal{H}^{\mathcal{L}}_{2r+6,r+11,r}=\mathcal{H}^{\mathcal{L}}_{20,18,7}$
is irreducible. By the observation, the only possibility is that all three
families $\cal{F}_{0},\mathcal{F}_{1},\mathcal{F}_{2}$ are contained in
another irreducible family of dimension strictly bigger than
$\dim\mathcal{F}_{1}$. (C): In what follows we argue that this is impossible
by showing that $\mathcal{F}_{2}$ is dense in a component
$\widetilde{\mathcal{F}}_{2}\subset\mathcal{G}^{7}_{20}$ and hence
$\dim\widetilde{\mathcal{F}}_{2}=33$.

Take a general $\cal{D}\in\cal{F}_{2}$ and set $\cal{E}=\cal{D}^{\vee}$. Let
$C_{\mathcal{E}}\in(10;4,3^{4})$; cf. (22). $C_{\mathcal{E}}\subset
S\subset\mathbb{P}^{4}$ is smooth where $S$ is a smooth del Pezzo surface. Let
$c\in\mathcal{H}^{\mathcal{L}}_{14,18,4}$ be the point corresponding to
$C_{\mathcal{E}}$ and let $\mathcal{H}$ be a component cotaining $c$. We
compute

$h^{0}({C_{\mathcal{E}}},N_{{C_{\mathcal{E}}}|\mathbb{P}^{4}})=\dim
T_{c}{\mathcal{H}}$

as follows.

We consider the standard exact sequence

$0\rightarrow N_{{C_{\mathcal{E}}}|S}\rightarrow
N_{{C_{\mathcal{E}}}|\mathbb{P}^{4}}\rightarrow
N_{{S|\mathbb{P}^{4}}_{|{C_{\mathcal{E}}}}}\rightarrow 0.$

Note that

$N_{C_{\mathcal{E}}|S}=\mathcal{O}_{S}({C_{\mathcal{E}}})_{|{C_{\mathcal{E}}}}=\mathcal{O}_{S}({C_{\mathcal{E}}})\otimes\mathcal{O}_{C_{\mathcal{E}}}.$

(i) Claim: $h^{0}({C_{\mathcal{E}}},N_{{C_{\mathcal{E}}}|S})=31$ and
$h^{1}({C_{\mathcal{E}}},N_{{C_{\mathcal{E}}}|S})=0$ The first part follows
from the exact sequence

$0\rightarrow\mathcal{O}_{S}\rightarrow\mathcal{O}_{S}({C_{\mathcal{E}}})\rightarrow\mathcal{O}_{S}({C_{\mathcal{E}}})\otimes\mathcal{O}_{C_{\mathcal{E}}}\rightarrow
0;$

a routine computation yields $h^{0}(S,\mathcal{O}_{S}({C_{\mathcal{E}}}))=32$
and since $S$ is rational, we have $h^{1}(S,\mathcal{O}_{S})=0$. Since
$\deg\mathcal{O}_{S}({C_{\mathcal{E}}})\otimes\mathcal{O}_{C_{\mathcal{E}}}={C_{\mathcal{E}}}^{2}=(10;4,3^{4})^{2}=48,$
$\mathcal{O}_{S}({C_{\mathcal{E}}})\otimes\mathcal{O}_{C_{\mathcal{E}}}$ is
non-special and hence $h^{1}({C_{\mathcal{E}}},N_{{C_{\mathcal{E}}}|S})=0$.

(ii) Claim:

$h^{0}({C_{\mathcal{E}}},N_{{S|\mathbb{P}^{4}}_{|{C_{\mathcal{E}}}}})=h^{0}({C_{\mathcal{E}}},N_{S|\mathbb{P}^{4}}\otimes\mathcal{O}_{C_{\mathcal{E}}})=26.$

Since $S$ is a complete intersection of two quadrics, we have

$N_{S|\mathbb{P}^{4}}=\mathcal{O}_{S}(2)\oplus\mathcal{O}_{S}(2),$

and hence