# Components of the Hilbert Scheme of smooth projective curves using ruled
surfaces II: existence of non-reduced components

Youngook Choi  Department of Mathematics Education, Yeungnam University, 280
Daehak-Ro,
Gyeongsan, Gyeongbuk 38541, Republic of Korea<EMAIL_ADDRESS>,  Hristo
Iliev  American University in Bulgaria, 2700 Blagoevgrad, Bulgaria, and
Institute of Mathematics and Informatics, Bulgarian Academy of Sciences,
1113 Sofia, Bulgaria<EMAIL_ADDRESS><EMAIL_ADDRESS>and  Seonja Kim
Department of Electronic Engineering, Chungwoon University, Sukgol-ro, Nam-gu,
Incheon 22100, Republic of Korea<EMAIL_ADDRESS>
(Date: July 21, 2022)

###### Abstract.

Let $\mathcal{I}_{d,g,r}$ be the union of irreducible components of the
Hilbert scheme whose general points represent smooth irreducible non-
degenerate curves of degree $d$ and genus $g$ in $\mathbb{P}^{r}$. Using a
family of curves found on ruled surfaces over smooth curves of genus $\gamma$,
we show that for $\gamma\geq 7$ and $g\geq 6\gamma+5$ the scheme
$\mathcal{I}_{2g-4\gamma+1,g,g-3\gamma+1}$ acquires a non-reduced component
$\mathcal{D}^{\prime}$ such that
$\operatorname{dim}T_{[X^{\prime}]}\mathcal{D}^{\prime}=\operatorname{dim}\mathcal{D}^{\prime}+1$
for a general point $[X^{\prime}]\in\mathcal{D}^{\prime}$.

###### Key words and phrases:

Hilbert scheme of curves, double covering, ruled surfaces

###### Key words and phrases:

Hilbert scheme of curves, Brill-Noether theory, double covering

###### 2000 Mathematics Subject Classification:

Primary 14C05; Secondary 14H10

###### 2000 Mathematics Subject Classification:

Primary 14C05; Secondary 14H10

The first author was supported by Basic Science Research Program through the
National Research Foundation of Korea(NRF) funded by the Ministry of
Education(2019R1I1A3A01055643). The third author was supported by by the
National Research Foundation of Korea(NRF) grant funded by the Korea
government(MSIT) (2022R1A2C1005977).

##  1\. Introduction

Denote by $\mathcal{I}_{d,g,r}$ the union of irreducible components of the
Hilbert scheme whose general points correspond to smooth irreducible non-
degenerate curves of degree $d$ and genus $g$ in $\mathbb{P}^{r}$. In general
its geometry can be fairly complicated as it might have multiple components
which can be of different dimension and also differ in regards to their
generic smoothness. Recall that the minimal dimension of a component is
$\lambda_{d,g,r}:=(r+1)d-(r-3)(g-1)$. This number is called the _expected
dimension_ of a component of $\mathcal{I}_{d,g,r}$. When the _Brill-Noether
number_ $\rho(d,g,r):=g-(r+1)(g-d+r)$ is non-negative the scheme
$\mathcal{I}_{d,g,r}$ possesses the unique component dominating the moduli
space of curves $\mathcal{M}_{g}$. It is known as the _distinguished
component_. It is generically smooth and has the expected dimension, [Ser84].
When $\rho(d,g,r)>0$ and $d\geq
2(1-\frac{3}{r+1})g+1+\frac{12}{r+1}=:\eta_{3}$ the scheme
$\mathcal{I}_{d,g,r}$ is irreducible, see [Kim01, Theorem 3.3] for details. If
we fix $g$ and $r$, and trace the change of geometry of $\mathcal{I}_{d,g,r}$
as the degree $d$ decreases from $\eta_{3}$, we get to observe interesting
phenomena: for $\eta_{4}:=2(1-\frac{4}{r+1})g+1+\frac{26}{r+1}<d\leq\eta_{3}$
the scheme $\mathcal{I}_{d,g,r}$ is irreducible if and only if
$3\nmid(2g-2-d)$; for $d\leq\eta_{4}$ the scheme $\mathcal{I}_{d,g,r}$ is
reducible, and in particular for $d\leq
2(1-\frac{8}{r+3})g+2+\frac{8}{r+3}=:\xi_{2}$ it acquires components that
parametrize curves that are double covers, see [CIK17, Proposition 3.2,
Theorem 4.3 and Theorem 4.4] and [CIK21, Theorem A]. Our paper [CIK21]
extended the result [CIK17, Theorem 4.3] as we showed that the corresponding
components are generically smooth. The main result of the present establishes
the existence of families of curves related to those from [CIK17, Theorem 4.4]
that give rise to non-reduced components of the Hilbert scheme. It is
contained in the next theorem.

######  Theorem B.

Assume that $g$ and $\gamma$ are integers such that $\gamma\geq 7$ and $g\geq
6\gamma+5$. Let

$d:=2g-4\gamma+2\quad\mbox{ and }\quad R:=g-3\gamma+2\,.$

Then the Hilbert scheme $\mathcal{I}_{d-1,g,R-1}$ possesses a _non-reduced_
component $\mathcal{D}^{\prime}_{d-1,g,R-1}$ such that

  1. (i)

$\operatorname{dim}\mathcal{D}^{\prime}_{d-1,g,R-1}=\lambda_{d-1,g,R-1}+(R-1)\gamma-2g+3=2g-\gamma-1+R^{2}$;

  2. (ii)

at a general point $[X^{\prime}]\in\mathcal{D}^{\prime}_{d-1,g,R-1}$ we have
$\operatorname{dim}T_{[X^{\prime}]}\mathcal{D}^{\prime}_{d-1,g,R-1}=\operatorname{dim}\mathcal{D}^{\prime}_{d-1,g,R-1}+1$;

  3. (iii)

a general point $[X^{\prime}]\in\mathcal{D}^{\prime}_{d-1,g,R-1}$ represents a
curve $X^{\prime}$ lying on a cone $F^{\prime}$ over a smooth curve
$Y^{\prime}$ of genus $\gamma$ and degree $g-2\gamma$ in $\mathbb{P}^{R-2}$
such that

    1. (iii.1)

$X^{\prime}\subset\mathbb{P}^{R-1}$ is linearly normal and passes through the
vertex $P^{\prime}$ of the cone $F^{\prime}$;

    2. (iii.2)

the projection from $P^{\prime}$ to the hyperplane in $\mathbb{P}^{R-1}$
containing $Y^{\prime}$ induces a morphism ${\varphi}^{\prime}:X^{\prime}\to
Y^{\prime}$ of degree two ;

    3. (iii.3)

$|\mathcal{O}_{X^{\prime}}(1)|\equiv|R_{{\varphi}^{\prime}}-\zeta|$ on
$X^{\prime}$, where $R_{{\varphi}^{\prime}}$ is the ramification divisor of
${\varphi}^{\prime}$ and $\zeta\in X^{\prime}$ is a point.

We remark that David Mumford showed in his famous paper on the pathologies of
Hilbert scheme [Mum62] that $\mathcal{I}_{14,24,3}$ is non-reduced. Since then
a considerable number of examples of non-reduced components of the Hilbert
scheme of space curves $\mathcal{I}_{d,g,3}$ has been produced, see [Ser82,
GP82, Kle87, Ell87, BKMR91, MDP96, Nas06, KO15, Dan17]. However, all of them
have relied on allocating a family of curves with desired invariants on a
surface in $\mathbb{P}^{3}$, which suggests that to construct non-reduced
components of $\mathcal{I}_{d,g,r}$ for $r\geq 4$ one needs to use a different
approach. Note that showing that a component of $\mathcal{I}_{d,g,r}$ is non-
reduced is equivalent to showing that a curve corresponding to a general point
of the component has a first order deformation that does not extend
infinitesimally. The last is harder to do when $r\geq 4$. To our knowledge,
the only example of a non-reduced component of $\mathcal{I}_{d,g,r}$ in the
case $r\geq 4$ has been the one given by Ciliberto, Lopez and Miranda, see
[CLM96, Theorem 4.11].

The primary objective of our study was to develop our understanding of the
geometry of components of $\mathcal{I}_{d,g,r}$, in particular, the geometry
of the families of curves related to those in [CIK17, Theorem 4.4]. To a large
extent this work builds upon our previous papers [CIK17] and [CIK21] and
should be regarded in their context. For this reason we consider integers
$\gamma\geq 7$ and $g\geq 6\gamma+5$ and fix the numbers $d$ and $R$
throughout the paper as

$d:=2g-4\gamma+2\quad\mbox{ and }\quad R:=g-3\gamma+2\,,$

exactly as they appeared in [CIK21, Theorem A] where we showed the existence
of a reduced irreducible component $\mathcal{D}_{d,g,R}$ of
$\mathcal{I}_{d,g,R}$ parametrizing curves on cones over smooth curves of
genus at least $10$ and degree $g-2\gamma+1$ in $\mathbb{P}^{r-1}$. The curves
parametrized by $\mathcal{D}^{\prime}_{d-1,g,R-1}$ in Theorem B arise as inner
projections of the former (this is the reason for ’prime’ in the notations),
which allows us to allocate the last on cones and to compute both the
dimension of the family and the dimension of the tangent space to
$\mathcal{I}_{d-1,g,R-1}$ at a general
$[X^{\prime}]\in\mathcal{D}^{\prime}_{d-1,g,R-1}$. We explain in details this
geometric relation between the two families in Section 2 together with some
preliminary facts. They motivate the rigorous construction and the proof of
Theorem B given in Section 3. It proceeds three steps: construction of the
family $\mathcal{F}^{\prime}$ with the desired invariants; computation of the
dimension of the tangent space to $\mathcal{D}^{\prime}_{d-1,g,R-1}$ at a
general point $[X^{\prime}]\in\mathcal{D}^{\prime}_{d-1,g,R-1}$, where we take
$\mathcal{D}^{\prime}_{d-1,g,R-1}$ to be the closure in
$\mathcal{I}_{d-1,g,R-1}$ of the set parametrizing $\mathcal{F}^{\prime}$;
showing that $\mathcal{D}^{\prime}_{d-1,g,R-1}$ is not properly contained in a
component of $\mathcal{I}_{d-1,g,R-1}$.

We consider schemes over $\mathbb{C}$. By _curve_ we understand a smooth
integral projective algebraic curve. Given a line bundle $L$ on a smooth
projective variety $X$ we denote by $|L|$ the complete linear series
$\mathbb{P}\left(H^{0}(X,L)\right)$ on $X$. When $X$ is an object of a family,
we denote by $[X]$ the corresponding point of the Hilbert scheme representing
the family. Oftentimes we abbreviate the notation of a line bundle
$L\otimes\mathcal{O}_{X}(D)$ to $L(D)$ for a divisor $D$ on $X$. We use $\sim$
to denote linear equivalence of divisors. For a smooth curve $C$ of genus $g$
we denote by $\operatorname{gon}(C)$ the gonality of $C$, that is, the minimal
$k$ for which there exists a pencil $g^{1}_{k}$ on $C$. Recall that if
$[C]\in\mathcal{M}_{g}$ is general then
$\operatorname{gon}(C)=\lfloor\frac{g+3}{2}\rfloor$. For all definitions and
properties of the objects not explicitly introduced in the paper refer to
[Har77] and [ACGH85]. Since this work is closely related to the results
obtained in our two previous papers [CIK17] and [CIK21], as we refer to them
often in what follows, we have opted to fix the values of $d$ as
$d=2g-4\gamma+2$ and $R$ as $R=g-3\gamma+2$ and to denote the component in
Theorem B as $\mathcal{D}^{\prime}_{d-1,g,R-1}$. We believe that this allows
the results from the three papers to be viewed and understood in the same
context.

##  2\. Motivation and preliminary results

In [CIK17, Theorem 4.3] we obtained a component $\mathcal{D}_{d,g,r}$ of
$\mathcal{I}_{d,g,r}$ under certain numerical assumptions about $d$, $g$ and
$r\leq R$. It was shown that a general point of $\mathcal{D}_{d,g,r}$
corresponds to a curve $X$ that is a double covering $\varphi:X\to Y$ of a
curve $Y$ of genus $\gamma$ as $\gamma=\frac{2g+2-d}{4}$ and the embedding
$X\hookrightarrow\mathbb{P}^{r}$ is given by a sub-series of the space of
global sections of
$\mathcal{O}_{X}(R_{\varphi})=\omega_{X}\otimes(\varphi^{\ast}\omega_{Y})^{\vee}$,
where $R_{\varphi}$ is the ramification divisor of $\varphi$. Similarly,
[CIK17, Theorem 4.4] showed the existence of a component
$\mathcal{D}^{\prime}_{d-1,g,r-1}$ of $\mathcal{I}_{d-1,g,r-1}$ whose general
point is also a double covering $\varphi^{\prime}:X^{\prime}\to Y^{\prime}$ of
a genus $\gamma$ curve but the embedding that time was given by a sub-series
of $|\mathcal{O}_{X^{\prime}}(R_{\varphi^{\prime}})(-x)|$ determined by the
ramification divisor $R_{\varphi^{\prime}}$ and an arbitrary point $x\in
X^{\prime}$. In order to uncover more of the geometry of the family of curves
parametrized by $\mathcal{D}_{d,g,R}\subset\mathcal{I}_{d,g,R}$, we
reconstructed it in [CIK21, Theorem A] as quadratic hypersurface sections of
cones in $\mathbb{P}^{R}$ over smooth curves of genus $\gamma\geq 10$ and
degree $g-2\gamma+1$ in $\mathbb{P}^{R-1}$. We remark that the curves
parametrized by $\mathcal{D}_{d,g,r}$ for $r<R$ were obtained as generic
projections to a general $\mathbb{P}^{r}\subset\mathbb{P}^{R}$, but in what
follows we consider only the case $r=R$. Since both coverings $\varphi:X\to Y$
and $\varphi^{\prime}:X^{\prime}\to Y^{\prime}$ were general, we can further
identify the curves parametrized by $\mathcal{D}^{\prime}_{d-1,g,R-1}$ as
inner projections of the curves parametrized by $\mathcal{D}_{d,g,R}$.
Precisely, let $X\hookrightarrow\mathbb{P}^{R}$ corresponds to a general point
of $\mathcal{D}_{d,g,R}$. By [CIK21, Theorem A] $X$ lies on a cone
$F\subset\mathbb{P}^{R}$ over a genus $\gamma$ and degree $g-2\gamma+1$ curve
$Y\subset\mathbb{P}^{R-1}$ as $X$ is cut on $F$ by a general quadratic
hypersurface in $\mathbb{P}^{R}$. Denote by $P$ the vertex of $F$. The
projection of $X$ from $P$ to the hyperplane $\mathbb{P}^{R-1}$ containing
$Y$, or equivalently the ruling of $F$, exhibits $X$ as double covering
$\varphi:X\to Y$. According to [CIK21]
$\mathcal{O}_{X}(1)=\mathcal{O}_{X}(R_{\varphi})=\omega_{X}\otimes(\varphi^{\ast}\omega_{Y})^{\vee}$.
Let $x\in X$ be a point and $H\subset\mathbb{P}^{R}$ be a general hyperplane.
Consider the linear projection

$\pi_{x}\,:\,X\dashrightarrow H$

with center $x$. Since $X$ doesn’t have trisecant lines it follows by [FOV99,
Proposition 8.4.3] that the rational map $\pi_{x}$ extends to an embedding of
$X$ in $H\cong\mathbb{P}^{R-1}$ as the degree of $X^{\prime}:=\pi_{x}(X)$ is
$\deg X^{\prime}=\deg X-1=d-1=2g-4\gamma+1$. Remark that the image of $x$ is
the point at which the tangent line to $X$ at $x$ meets $H$. Further, abusing
notation, we consider also the linear projection

$\pi_{x}\,:\,F\dashrightarrow H$

with center $x$. It maps the cone $F$ to a cone
$F^{\prime}\subset\mathbb{P}^{R-1}$ as the line passing through $x$ and the
vertex $P$ of $F$ is contracted to the vertex $P^{\prime}$ of $F^{\prime}$,
see Figure 1.

$Y$$\mathbb{P}^{R-1}$$\mathbb{P}^{R}$$F$$X$$P$$x$$Y^{\prime}$$\mathbb{P}^{R-2}$$H\cong\mathbb{P}^{R-1}$$F^{\prime}$$X^{\prime}$$P^{\prime}$$\pi_{x}$
Figure 1. The inner projection of $X\subset F\subset\mathbb{P}^{R}$ from a
point $x\in X$ to a general hyperplane $H$ in $\mathbb{P}^{R}$. The image of
$F$ is the the cone $F^{\prime}$ and the image of $X$ is the curve
$X^{\prime}$ on $F^{\prime}$ passing through the vertex of $F^{\prime}$.

######  Remark 1.

It is easy to see that under the linear map $\pi_{x}$:

  1. (1)

a line of the ruling of $F$ that doesn’t pass through $x$ is mapped to a line
from the ruling of $F^{\prime}$;

  2. (2)

a hyperplane section of $F$ that doesn’t pass through $x$ is embedded in $H$
as a curve of degree $g-2\gamma+1$ on $F^{\prime}$ that passes through
$P^{\prime}$;

  3. (3)

a hyperplane section of $F$ that passes through $x$ is mapped isomorphically
to a hyperplane section of $F^{\prime}$ as it is properly contained in the
intersection $\mathbb{P}^{R-2}$ of $H$ and the hyperplane that cuts $Y$ on
$F$.

Since the line from the ruling of $F$ passing through $x$ meets $X$ in a
second point, it follows that the isomorphic image $X^{\prime}$ of $X$ passes
through $P^{\prime}$. Let us remark explicitly that if $Y$ is a general
hyperplane section of $F$ passing through $x$, then its image $Y^{\prime}$
will be a curve of genus $\gamma$ and degree $g-2\gamma$ that is properly
contained in $\mathbb{P}^{R-2}$. The projection of $X^{\prime}$ from
$P^{\prime}$ to the hyperplane in $H\cong\mathbb{P}^{R-1}$ that cuts
$Y^{\prime}$ on $F^{\prime}$ gives a double covering morphism
$\varphi^{\prime}:X^{\prime}\to Y^{\prime}$. Due to $X^{\prime}$ being an
inner projection of $X$ with center $x$ we have that the embedding
$X^{\prime}\hookrightarrow H\cong\mathbb{P}^{R-1}$ is induced by the line
bundle

$\mathcal{O}_{X}(1)(-x)\cong\mathcal{O}_{X}(R_{\varphi}-x)\cong\omega_{X}\otimes(\varphi^{\ast}\omega_{Y})^{\vee}(-x)$

as in [CIK17, Theorem 4.4]. This geometric picture hints that we can
reconstruct the family from [CIK17, Theorem 4.4] as a family of curves on
elementary transformations of the ruled surfaces used in [CIK21, Theorem A],
see Figure 2 and Figure 3 in section 3.

Before proceeding with the rigorous construction of the component
$\mathcal{D}^{\prime}_{d-1,g,R-1}$ and the proof of Theorem B we formulate and
prove the following auxiliary result which will be used in the calculation of
$h^{0}(X^{\prime},N_{X^{\prime}/\mathbb{P}^{R-1}})$.

######  Proposition 2.

Let $C$ be a non-degenerate smooth integral curve of genus $g$ in
$\mathbb{P}^{r}$, $r\geq 3$. Let $H$ be a hyperplane in $\mathbb{P}^{r}$ and
$x$ be a point on $C$. Suppose that the inner projection $\phi\,:\,C\to
H\cong\mathbb{P}^{r-1}$ with center $x$ maps $C$ to a non-degenerate smooth
integral curve $D$ of genus $\gamma$ in $H$. Denote by $R_{\phi}$ the
ramification divisor of $\phi$. Then

(1)
$0\to\mathcal{O}_{C}(R_{\phi})\otimes\mathcal{O}_{C}(1)\otimes\mathcal{O}_{C}(2x)\to
N_{C/\mathbb{P}^{r}}\to\phi^{\ast}N_{D/\mathbb{P}^{r-1}}\otimes\mathcal{O}_{C}(x)\to
0\,,$

where $N_{C/\mathbb{P}^{r}}$ is the normal bundle of $C$ in $\mathbb{P}^{r}$
and $N_{D/\mathbb{P}^{r-1}}$ is the normal bundle of $D$ in
$H\cong\mathbb{P}^{r-1}$.

###### Proof of Proposition 2.

Let $\sigma\,:\,\widetilde{\mathbb{P}^{r}}(x)\to\mathbb{P}^{r}$ be the blow-up
of $\mathbb{P}^{r}$ at $x$ and let $E$ be the exceptional divisor
$E=\sigma^{-1}(x)\cong\mathbb{P}^{r-1}$. The tautological line bundle
$\mathcal{O}_{\widetilde{\mathbb{P}}^{r}(x)}(1)$ on
$\widetilde{\mathbb{P}}^{r}(x)$ is isomorphic to
$\sigma^{\ast}\mathcal{O}_{\mathbb{P}^{r}}(1)$. Since $C$ is smooth, the
strict transform $\tilde{C}$ of $C$ in $\widetilde{\mathbb{P}^{r}}(x)$ is
isomorphic to $C$ and
${\mathcal{O}_{\widetilde{\mathbb{P}}^{r}(x)}(1)}{|_{\tilde{C}}}\cong\sigma^{\ast}\mathcal{O}_{C}(1)$.
Let $\varepsilon\,:\,\widetilde{\mathbb{P}^{r}}(x)\to\mathbb{P}^{r-1}$ be the
morphism defined by the linear series
$|\sigma^{\ast}\mathcal{O}_{\mathbb{P}^{r}}(1)-E|$. Then $\phi\,:\,C\to D$ can
be regarded as the restriction of $\varepsilon$ on $\tilde{C}$. Since
$\tilde{C}$ meets $E$ at a single point, it follows that
$\phi^{\ast}\mathcal{O}_{D}(1)\cong\mathcal{O}_{C}(1)\otimes\mathcal{O}_{C}(-x)$
on $C$.

Consider the Euler sequences

$0\to\mathcal{O}_{C}\to\oplus^{r+1}_{1}\mathcal{O}_{C}(1)\to
T_{\mathbb{P}^{r}|_{C}}\to 0\,$

and

$0\to\mathcal{O}_{D}\to\oplus^{r}_{1}\mathcal{O}_{D}(1)\to
T_{\mathbb{P}^{r-1}|_{D}}\to 0\,.$

Pulling back the second exact sequence to $C$ via $\phi^{\ast}$ we obtain

$\begin{array}[]{ccccccccccccccccccccccc}&&&&0&&0&&\\\\[3.87498pt]
&&&&\downarrow&&\downarrow&&\\\\[3.87498pt]
&&0&&\mathcal{O}_{C}(1)&&\ker(\alpha)&&\\\\[3.87498pt]
&&\downarrow&&\downarrow&&\downarrow&\\\\[3.87498pt]
0&\rightarrow&\mathcal{O}_{C}&\rightarrow&\oplus^{r+1}_{1}\mathcal{O}_{C}(1)&\rightarrow&T_{\mathbb{P}^{r}|_{C}}&\
\rightarrow&0\\\\[3.87498pt]
&&\downarrow&&\downarrow&&\downarrow\alpha&\\\\[3.87498pt]
0&\rightarrow&\mathcal{O}_{C}(x)&\rightarrow&{\phi}^{\ast}\left(\oplus^{r}_{1}\mathcal{O}_{D}(1)\right)\otimes\mathcal{O}_{C}(x)&\rightarrow&{\phi}^{\ast}\left(T_{\mathbb{P}^{r-1}|_{D}}\right)\otimes\mathcal{O}_{C}(x)&\rightarrow&0\\\\[3.87498pt]
&&\downarrow&&\downarrow&&\downarrow&\\\\[3.87498pt]
&&\mathcal{O}_{x}&&0&&0&&\\\\[3.87498pt] &&\downarrow&&&&&\\\\[3.87498pt]
&&0&&&&&\\\\[3.87498pt] \end{array}$

Chasing the above diagram we obtain by the Snake Lemma that
$\ker(\alpha)=\mathcal{O}_{C}(1)\otimes\mathcal{O}_{C}(x)$, whence

$0\to\mathcal{O}_{C}(1)\otimes\mathcal{O}_{C}(x)\to
T_{\mathbb{P}^{r}|_{C}}\to{\phi}^{\ast}\left(T_{\mathbb{P}^{r-1}|_{D}}\right)\otimes\mathcal{O}_{C}(x)\to
0\,.$

Further, using the canonical sequence for $N_{C/\mathbb{P}^{r}}$

$0\to T_{C}\to{T_{\mathbb{P}^{r}}}_{|_{C}}\to N_{C/\mathbb{P}^{r}}\to 0$

and the one for $N_{D/\mathbb{P}^{r-1}}$ tensored by $\mathcal{O}_{C}(x)$

$0\to{\phi}^{\ast}(T_{D})\otimes\mathcal{O}_{C}(x)\to{\phi}^{\ast}({T_{\mathbb{P}^{r-1}}}_{|_{D}})\otimes\mathcal{O}_{C}(x)\to{\phi}^{\ast}(N_{D/\mathbb{P}^{r-1}})\otimes\mathcal{O}_{C}(x)\to
0\,,$

we get the commutative diagram

$\begin{array}[]{ccccccccccccccccccccccc}&&&&0&&0&&\\\\[4.30554pt]
&&&&\downarrow&&\downarrow&&\\\\[4.30554pt]
&&0&&\mathcal{O}_{C}(1)\otimes\mathcal{O}_{C}(x)&&\ker(\beta)&&\\\\[4.30554pt]
&&\downarrow&&\downarrow&&\downarrow&\\\\[4.30554pt]
0&\rightarrow&T_{C}&\rightarrow&T_{\mathbb{P}^{r}|_{C}}&\rightarrow&N_{C/\mathbb{P}^{r}}&\rightarrow&0&\\\\[4.30554pt]
&&\downarrow&&\downarrow&&\downarrow\beta&\\\\[4.30554pt]
0&\to&{\phi}^{\ast}(T_{D})\otimes\mathcal{O}_{C}(x)&\to&{\phi}^{\ast}({T_{\mathbb{P}^{r-1}}}_{|_{D}})\otimes\mathcal{O}_{C}(x)&\to&{\phi}^{\ast}(N_{D/\mathbb{P}^{r-1}})\otimes\mathcal{O}_{C}(x)&\to&0\\\\[4.30554pt]
&&\downarrow&&\downarrow&&\downarrow&\\\\[4.30554pt]
&&\mathcal{O}_{R_{\phi}+x}&&0&&0&&\\\\[4.30554pt]
&&\downarrow&&&&&\\\\[4.30554pt] &&0&&&&&\\\\[4.30554pt] \end{array}$

Similarly as before,
$\ker\beta=\mathcal{O}_{C}(1)\otimes\mathcal{O}_{C}(R_{\phi}+2x)$, from where
we obtain the short exact sequence

$0\to\mathcal{O}_{C}(1)\otimes\mathcal{O}_{C}(R_{\phi}+2x)\to
N_{C/\mathbb{P}^{r}}\to{\phi}^{\ast}N_{D/\mathbb{P}^{r-1}}\otimes\mathcal{O}_{C}(x)\to
0\,.$

∎

######  Remark 3.

When $\phi$ is an isomorphism the claim is contained in [Ein87, Lemma 4], see
also [Wal94, Lemma 6] and [FS19, Proposition 6]. We will use Proposition 2 in
the proof of Theorem B in the case of $\phi$ being a double covering.

##  3\. Proof of Theorem B

We recall the basic numerical assumptions in the theorem: $\gamma\geq 7$,
$g\geq 6\gamma+5$, $d=2g-4\gamma+2$ and $R=g-3\gamma+2$. The proof proceeds in
three main steps:

  * Step I.

We construct a family $\mathcal{F}^{\prime}$ of curves satisfying the
condition $(iii)$ of Theorem B, then we consider the closure
$\mathcal{D}^{\prime}_{d-1,g,R-1}$ of the subset of $\mathcal{I}_{d-1,g,R-1}$
parametrizing the family $\mathcal{F}^{\prime}$ and show that

$\operatorname{dim}\mathcal{D}^{\prime}_{d-1,g,R-1}=2g-\gamma-1+R^{2}\,.$

  * Step II.

For a general curve $X$ from the family $\mathcal{F}^{\prime}$ we show that

$\operatorname{dim}T_{[X]}\mathcal{D}^{\prime}_{d-1,g,R-1}=h^{0}(X,N_{X/\mathbb{P}^{R-1}})=2g-\gamma+R^{2}=\operatorname{dim}\mathcal{D}^{\prime}_{d-1,g,R-1}+1\,.$

  * Step III.

We show that there is no component of $\mathcal{I}_{d-1,g,R-1}$ that properly
contains $\mathcal{D}^{\prime}_{d-1,g,R-1}$.

The main claim in Theorem B is obtained upon the completion of those three
steps. The characterization of the curves parametrized by
$\mathcal{D}^{\prime}_{d-1,g,R-1}$ as in item (iii) of Theorem B is found in
Step I and Step II.

Step I. Let $\Gamma\in\mathcal{M}_{\gamma}$ be a general curve of genus
$\gamma$ and $E^{\prime}$ be a divisor of degree $g-2\gamma$ on $\Gamma$. Let
$q\in\Gamma$ be a point and $E:=E^{\prime}+q$ be the divisor of degree
$g-2\gamma+1$ on $\Gamma$. Consider the ruled surface
$S^{\prime}:=\mathbb{P}(\mathcal{O}_{\Gamma}\oplus\mathcal{O}_{\Gamma}(-E^{\prime}))$
with natural projection $f^{\prime}:S^{\prime}\to\Gamma$ and the ruled surface
$S:=\mathbb{P}(\mathcal{O}_{\Gamma}\oplus\mathcal{O}_{\Gamma}(-E))$ with
natural projection $f:S\to\Gamma$. Denote by $\Gamma^{\prime}_{0}$ the section
of minimal self-intersection on $S^{\prime}$ and by $\Gamma_{0}$ the section
of minimal self-intersection on $S$. Recall that
${\Gamma^{\prime}_{0}}^{2}=\deg\mathcal{O}_{\Gamma}(-E^{\prime})=-g+2\gamma$
and ${\Gamma_{0}}^{2}=\deg\mathcal{O}_{\Gamma}(-E)=-g+2\gamma-1$. As it is
known about ruled surfaces,
$\operatorname{Pic}(S^{\prime})\cong\mathbb{Z}[\Gamma^{\prime}_{0}]\oplus{f^{\prime}}^{\ast}(\operatorname{Pic}(\Gamma))$
and
$\operatorname{Pic}(S)\cong\mathbb{Z}[\Gamma_{0}]\oplus{f}^{\ast}(\operatorname{Pic}(\Gamma))$.
Further, for a divisor $D$ on $\Gamma$ we will denote by
$D\mathfrak{f}^{\prime}$ the divisor ${f^{\prime}}^{\ast}(D)$ on $S^{\prime}$
and by $D\mathfrak{f}$ the divisor ${f}^{\ast}(D)$ on $S$. Remark that the
ruled surfaces $S^{\prime}$ and $S$ are elementary transformations of one
another. Namely, if $x$ is a point in the fiber of $S$ over $q\in\Gamma$ such
that $x\notin\Gamma_{0}$, then the composition $\nu$ of blowing up
$\sigma:S_{x}\to S$ of $S$ at ${x}$ followed by blowing down $\varepsilon$ of
$S_{x}$ at the strict transform $\sigma^{-1}(q\mathfrak{f})$ of
$q\mathfrak{f}$ yields the elementary transformation $\operatorname{elm}_{x}$
of $S$ at $x$ and $\operatorname{elm}_{x}S\equiv S^{\prime}$, see [FP05b].
Note also that $\nu$ contracts $q\mathfrak{f}$ to the unique point on the
intersection of $\Gamma^{\prime}_{0}$ and $q\mathfrak{f}^{\prime}$.
Conversely, if $x^{\prime}=\Gamma^{\prime}_{0}\cap q\mathfrak{f}^{\prime}$,
then the blow up $\varepsilon:S^{\prime}_{x^{\prime}}\equiv S_{x}\to
S^{\prime}$ of $S^{\prime}$ at ${x}^{\prime}$ followed by the blow down
$\sigma$ of $S^{\prime}_{x^{\prime}}$ at the strict transform
$\varepsilon^{-1}(q\mathfrak{f}^{\prime})$ of $q\mathfrak{f}^{\prime}$ gives
$\operatorname{elm}_{x^{\prime}}S^{\prime}=\mathbb{P}(\mathcal{O}_{\Gamma}\oplus\mathcal{O}_{\Gamma}(-E^{\prime}-q))=S$,
see Figure 2.

$x$$q\mathfrak{f}$${\Gamma}_{0}$$C$$\Lambda$$S=\mathbb{P}(\mathcal{O}_{\Gamma}\oplus\mathcal{O}_{\Gamma}(-E))$$S^{\prime}=\mathbb{P}(\mathcal{O}_{\Gamma}\oplus\mathcal{O}_{\Gamma}(-E^{\prime}))$$x^{\prime}$$q{\mathfrak{f}^{\prime}}$${\Gamma}^{\prime}_{0}$$C^{\prime}$$\Lambda^{\prime}$$\nu=\operatorname{elm}_{x}$$\nu^{-1}=\operatorname{elm}_{x^{\prime}}$$\operatorname{Bl}_{x}S=\operatorname{Bl}_{x^{\prime}}{S}^{\prime}$$\sigma$$\varepsilon$
Figure 2.  $S=\mathbb{P}(\mathcal{O}_{\Gamma}\oplus\mathcal{O}_{\Gamma}(-E))$
and
$S^{\prime}=\mathbb{P}(\mathcal{O}_{\Gamma}\oplus\mathcal{O}_{\Gamma}(-{E}^{\prime}))$,
$E=E^{\prime}+q$.

Let $C\in|2\Gamma_{0}+2E\mathfrak{f}|$ be general. Such curve $C$ meets
$q\mathfrak{f}$ in two points. To avoid complications with notations and to
use the setup developed above we denote one of them as $x$. As it is explained
in [FP05b], if $\Lambda$ is general in the sub-system of
$|\Gamma_{0}+(E^{\prime}+q)\mathfrak{f}|$ of divisors containing $x$, then the
strict transform $\Lambda^{\prime}$ of $\Lambda$ will be general in
$|{\Gamma}^{\prime}_{0}+E^{\prime}\mathfrak{f}^{\prime}|$. Also, the strict
transform $C^{\prime}$ of $C$ will be general in
$|2{\Gamma}^{\prime}_{0}+(2E^{\prime}+q)\mathfrak{f}^{\prime}|$. Note that it
follows by [FP05b, Proposition 28] that $x^{\prime}$ is the unique based point
of $|2{\Gamma}^{\prime}_{0}+(2E^{\prime}+q)\mathfrak{f}^{\prime}|$ on
$S^{\prime}$ while a general element of that linear series is a smooth
integral curve. Conversely, if
$\Lambda^{\prime}\in|{\Gamma}^{\prime}_{0}+E^{\prime}\mathfrak{f}^{\prime}|$
is general, then its image $\nu^{-1}(\Lambda^{\prime})$ will be general in the
sub-series of $|\Gamma_{0}+({E}^{\prime}+q)\mathfrak{f}|$ on $S$ of divisors
vanishing at $\nu^{-1}(q\mathfrak{f}^{\prime})=x$. Similarly, a general
$C^{\prime}\in|2{\Gamma}^{\prime}_{0}+(2E^{\prime}+q)\mathfrak{f}^{\prime}|$
necessarily contains $x^{\prime}$ and its strict transform
$C:=\overline{\nu^{-1}(C^{\prime}\setminus\\{x^{\prime}\\})}$ will be general
in the sub-series of $|2\Gamma_{0}+2E\mathfrak{f}|$ on $S$ of divisors
vanishing at $x$, see Figure 2.

Before proceeding with the description of $\mathcal{D}^{\prime}_{d-1,g,R-1}$
let’s remark that due to our numerical assumptions and reasons similar to
those in [CIK21, Proposition 1], the morphism
$\Psi^{\prime}\,:\,S^{\prime}\to\mathbb{P}^{R-1}$ induced by the linear system
$|\Gamma^{\prime}_{0}+E^{\prime}\mathfrak{f}^{\prime}|$ on $S^{\prime}$ maps
$S^{\prime}$ to a cone $F^{\prime}\subset\mathbb{P}^{R-1}$ and a general
$C^{\prime}\in|2\Gamma^{\prime}_{0}+(2E^{\prime}+q)\mathfrak{f}^{\prime}|$ to
a curve $X^{\prime}\subset F^{\prime}$ of genus $g$ and degree
$d-1=2g-4\gamma+1$. The morphism $\Psi^{\prime}$ is isomorphism away from
$\Gamma^{\prime}_{0}$ and contracts the last to the vertex, say $P^{\prime}$,
of $F^{\prime}$. This implies that $X^{\prime}$ passes through $P^{\prime}$.
The hyperplane sections of $F^{\prime}$ are the images of curves from
$|\Gamma^{\prime}_{0}+E^{\prime}\mathfrak{f}^{\prime}|$, therefore we have
that $F^{\prime}$ is cone over a genus $\gamma$ curve, say $Y^{\prime}$, of
degree $\deg
Y^{\prime}=(\Gamma^{\prime}_{0}+E^{\prime}\mathfrak{f}^{\prime})\cdot(\Gamma^{\prime}_{0}+E^{\prime}\mathfrak{f}^{\prime})=-\deg
E^{\prime}+2\deg E^{\prime}=g-2\gamma$. Since

$\displaystyle
h^{0}(S^{\prime},\mathcal{O}_{S^{\prime}}(2{\Gamma}^{\prime}_{0}+(2E^{\prime}+q){\mathfrak{f}^{\prime}}))$
$\displaystyle=\sum^{2}_{k=0}h^{0}(\Gamma,\mathcal{O}_{\Gamma}((2E^{\prime}+q)-kE^{\prime}))=3g-8\gamma+5\,,$

and
$h^{0}(S^{\prime},\mathcal{O}_{S^{\prime}}({\Gamma}^{\prime}_{0}+E^{\prime}{\mathfrak{f}^{\prime}}))=R$,
it follows easily that the embedding
$X^{\prime}\hookrightarrow\mathbb{P}^{R-1}$ is linearly normal as well as
$Y^{\prime}\hookrightarrow\mathbb{P}^{R-2}$. Note that this proves (iii.1)
upon completion of all the three steps in the proof.

Define $\mathcal{F}^{\prime}$ to be the family of curves that are images of
the divisors from the linear series
$|2{\Gamma}^{\prime}_{0}+(2E^{\prime}+q){\mathfrak{f}^{\prime}}|$ on
$S^{\prime}=\mathbb{P}(\mathcal{O}_{\Gamma}\oplus\mathcal{O}_{\Gamma}(-E^{\prime}))$
under the morphism $\Psi^{\prime}$, defined as above, obtained by varying
$\Gamma$ in $\mathcal{M}_{\gamma}$, running $E^{\prime}$ through the set of
all divisors of degree $g-2\gamma$ on $\Gamma$ and $q\in\Gamma$. For the
dimension of $\mathcal{F}^{\prime}$ we have

  * $\operatorname{dim}\mathcal{F^{\prime}}=$

    * $+$

$3\gamma-3$ : number of parameters of curves $\Gamma\in\mathcal{M}_{\gamma}$

    * $+$

$\gamma$ : number of parameters of line bundles
$\mathcal{O}_{\Gamma}(E^{\prime})\in\operatorname{Pic}(\Gamma)$ of degree
$g-2\gamma\geq 4\gamma+5$ necessary to fix the geometrically ruled surface
$\mathbb{P}(\mathcal{O}_{\Gamma}\oplus\mathcal{O}_{\Gamma}(-E^{\prime}))$

    * $+$

$R^{2}-1=\operatorname{dim}(\operatorname{Aut}(\mathbb{P}^{R-1}))$

    * $+$

$1$ : number of parameters necessary to fix $q\in\Gamma$

    * $-$

$((g-2\gamma)-\gamma+2)=\operatorname{dim}G_{F^{\prime}}$, where
$G_{F^{\prime}}$ is the subgroup of $\operatorname{Aut}(\mathbb{P}^{R-1})$
fixing the scroll $F^{\prime}$, see [CCFM09, Lemma 6.4, p. 148]

    * $+$

$3g-8\gamma+4=\operatorname{dim}|2{\Gamma}^{\prime}_{0}+(2E^{\prime}+q){\mathfrak{f}^{\prime}}|$
: number of parameters to choose a curve in the linear equivalence class of
$2{\Gamma}^{\prime}_{0}+(2E^{\prime}+q){\mathfrak{f}^{\prime}}$ on
$S^{\prime}$.

Define $\mathcal{D}^{\prime}_{d-1,g,R-1}$ as the closure in
$\mathcal{I}_{d-1,g,R-1}$ of the set parametrizing $\mathcal{F}^{\prime}$.
Accounting the above numbers we get

$\operatorname{dim}\mathcal{D}^{\prime}_{d-1,g,R-1}=\operatorname{dim}\mathcal{F}^{\prime}=2g-\gamma-1+R^{2}\,.$

This completes Step I.

Step II. We preserve all notations introduced in Step I. Before proceeding to
the main computation in the second step in the proof of Theorem B, we need to
complete the geometric picture that relates the family $\mathcal{F}$
parametrized by the component $\mathcal{D}_{d,g,R}$ from [CIK21, Theorem A]
and the family $\mathcal{F}^{\prime}$. In terms of our notations, the curves
from $\mathcal{F}$ are the images of the divisors
$|2\Gamma_{0}+2E\mathfrak{f}|$ by the morphism $\Psi$ determined by the linear
series $|\Gamma_{0}+(E^{\prime}+q)\mathfrak{f}|$ on $S$. Note that such a
curve is cut by a quadratic hypersurface of the cone
$F:=\Psi(S)\subset\mathbb{P}^{R}$ over a curve of genus $\gamma$ and degree
$g-2\gamma+1$ in $\mathbb{P}^{R-1}$. By [FP05b, Lemma 45] we have that the
linear system $|\Gamma^{\prime}_{0}+E^{\prime}\mathfrak{f}^{\prime}|$ on
$S^{\prime}$ is determined precisely by the linear system consisting of the
global sections of $|\Gamma_{0}+E\mathfrak{f}|$ that vanish at $x$. In
particular, the cone $F^{\prime}$ can be seen as a linear projection of the
cone $F$ with center $\Psi(x)$ to a hyperplane $H$ in $\mathbb{P}^{R}$. For a
general $X^{\prime}\in\mathcal{F}^{\prime}$ there exists
$C^{\prime}\in|2{\Gamma}^{\prime}_{0}+(2E^{\prime}+q){\mathfrak{f}^{\prime}}|$
such that $\Psi^{\prime}(C^{\prime})=X^{\prime}$, and the above implies that
such a curve $X^{\prime}$ is the image of the projection with center $\Psi(x)$
of $X:=\Psi(C)$, where $C$ is the strict transform
$\overline{\nu^{-1}(C^{\prime}\setminus\\{x^{\prime}\\})}$ of $C^{\prime}$.
That is, the curves $X^{\prime}\subset\mathbb{P}^{R-1}$ from
$\mathcal{F}^{\prime}$ can be regarded as inner projections of the curves
$X\subset\mathbb{P}^{R}$ from the family $\mathcal{F}$. Moreover, if
$Y^{\prime}$ is a hyperplane section of $F^{\prime}$ then there is a curve
$\Lambda^{\prime}\in|{\Gamma}^{\prime}_{0}+E^{\prime}{\mathfrak{f}^{\prime}}|$
such that $Y^{\prime}=\Psi^{\prime}(\Lambda^{\prime})$. For this
$\Lambda^{\prime}$ we have that its strict transform
$\Lambda=\overline{\nu^{-1}(\Lambda^{\prime}\setminus\\{x^{\prime}\\})}$
contains $x$ and further, the projection of $Y:=\Psi(\Lambda)$ with center
$\Psi(x)$ to $H\cong\mathbb{P}^{R-1}$ is just $Y^{\prime}$, see Figure 3.

${\operatorname{Bl}_{x}S\equiv\operatorname{Bl}_{x^{\prime}}S^{\prime}}$${S}$${S^{\prime}}$${\mathbb{P}^{R}\supset
F\supset X\phantom{.}}$${\phantom{X.}\mathbb{P}^{R-1}\supset
Y}$${\phantom{X.}X^{\prime}\subset
F^{\prime}\subset\mathbb{P}^{R-1}}$${Y^{\prime}\subset\mathbb{P}^{R-2}\phantom{X.}}$$\scriptstyle{\sigma}$$\scriptstyle{\varepsilon}$$\scriptstyle{\nu=\operatorname{elm}_{x}}$$\scriptstyle{\nu^{-1}=\operatorname{elm}_{x^{\prime}}}$$\scriptstyle{\pi_{{\Psi(x)}}}$$\scriptstyle{\varphi}$$\scriptstyle{{\varphi}^{\prime}}$
Figure 3.  $F^{\prime}={\Psi}^{\prime}(S^{\prime})\subset\mathbb{P}^{R-1}$ is
the inner projection $\pi_{\Psi(x)}:F\to\mathbb{P}^{R-1}$ of
$F=\Psi(S)\subset\mathbb{P}^{R}$.

The ruling of $F^{\prime}$, or equivalently the projection with center
$P^{\prime}$ to the hyperplane in $H\cong\mathbb{P}^{R-1}$ that cuts
$Y^{\prime}$, yields the double covering morphism
$\varphi^{\prime}:X^{\prime}\to Y^{\prime}$. For this reason

${\varphi^{\prime}}^{\ast}\mathcal{O}_{Y^{\prime}}(1)\cong\mathcal{O}_{X^{\prime}}(1)\otimes\mathcal{O}_{X^{\prime}}(-P^{\prime})\,.$

If $x^{\prime\prime}$ is the second point, apart from $x^{\prime}$, in which
$q\mathfrak{f}^{\prime}$ meets $C^{\prime}$ and
$\zeta:=\Psi^{\prime}(x^{\prime\prime})$, then it can be easily seen that

$\mathcal{O}_{X^{\prime}}(1)\cong\mathcal{O}_{X^{\prime}}(R_{\varphi^{\prime}}-\zeta)\,,$

where $R_{\varphi^{\prime}}$ is the ramification divisor of
$\varphi^{\prime}$. This gives items (iii.2) and (iii.3) from Theorem B.

Now we compute $h^{0}(X^{\prime},N_{X^{\prime}/\mathbb{P}^{R-1}})$ for a
general $X^{\prime}$ from the family $\mathcal{F}^{\prime}$. We preserve all
notions introduced in Step I. As it was obtained, the double covering morphism
$\varphi^{\prime}$ is a linear projection with center $P^{\prime}$. Applying
Proposition 2 we get the short exact sequence

(2)
$0\to\mathcal{O}_{X^{\prime}}(1)\otimes\mathcal{O}_{X^{\prime}}(R_{{\varphi}^{\prime}}+2P^{\prime})\to
N_{X^{\prime}/\mathbb{P}^{R-1}}\to{{\varphi}^{\prime}}^{\ast}N_{Y^{\prime}/\mathbb{P}^{R-2}}\otimes\mathcal{O}_{X^{\prime}}(P^{\prime})\to
0\,,$

Remark that the Hilbert scheme $\mathcal{I}_{g-2\gamma,\gamma,g-3\gamma}$
generically smooth and irreducible due to $g-2\gamma\geq 4\gamma+5$ and
[Ein87, Theorem 8]. Also,
$\operatorname{dim}\mathcal{I}_{g-2\gamma,\gamma,g-3\gamma}=\lambda_{g-2\gamma,\gamma,R-2}=(g-2\gamma)(R-1)-(R-5)(\gamma-1)$.
The curve $Y^{\prime}$ corresponds to a general point of
$\mathcal{I}_{g-2\gamma,\gamma,g-3\gamma}$ because
$Y^{\prime}\cong\Gamma\in\mathcal{M}_{\gamma}$ is general and $\deg
Y^{\prime}=g-2\gamma\geq 4\gamma+5$ according to the assumptions in the
theorem. Therefore

$h^{0}(Y^{\prime},N_{Y^{\prime}/\mathbb{P}^{R-2}})=(g-2\gamma)(R-1)-(R-5)(\gamma-1)\,.$

Since $\deg R_{{\varphi}^{\prime}}=2g-4\gamma+2$ and
$\deg\mathcal{O}_{X^{\prime}}(1)=d-1=2g-4\gamma+1$, the line bundle
$\mathcal{O}_{X^{\prime}}(1)\otimes\mathcal{O}_{X^{\prime}}(R_{{\varphi}^{\prime}}+2P^{\prime})$
on $X^{\prime}$ has degree $4g-8\gamma+5\geq 2g+4\gamma+15$. Hence it is non-
special and by Riemann-Roch

$h^{0}(X^{\prime},\mathcal{O}_{X^{\prime}}(1)\otimes\mathcal{O}_{X^{\prime}}(R_{{\varphi}^{\prime}}+2P^{\prime}))=3g-8\gamma+6\,.$

For the computation of
$h^{0}(X^{\prime},{{\varphi}^{\prime}}^{\ast}N_{Y^{\prime}/\mathbb{P}^{R-2}}\otimes\mathcal{O}_{X^{\prime}}(P^{\prime}))$
we use that

$\displaystyle
h^{0}(X^{\prime},{{\varphi}^{\prime}}^{\ast}N_{Y^{\prime}/\mathbb{P}^{R-2}}\otimes\mathcal{O}_{X^{\prime}}(P^{\prime}))$
$\displaystyle=h^{0}(Y^{\prime},{\varphi}^{\prime}_{\ast}(\,{{\varphi}^{\prime}}^{\ast}N_{Y^{\prime}/\mathbb{P}^{R-2}}\otimes\mathcal{O}_{X^{\prime}}(P^{\prime})\,))$
$\displaystyle=h^{0}(Y^{\prime},N_{Y^{\prime}/\mathbb{P}^{R-2}}\otimes{\varphi}^{\prime}_{\ast}\mathcal{O}_{X^{\prime}}(P^{\prime}))\,.$

Claim.
${\varphi}^{\prime}_{\ast}\mathcal{O}_{X^{\prime}}(P^{\prime})=\mathcal{O}_{Y^{\prime}}\oplus\mathcal{O}_{Y^{\prime}}(-1)$.

###### Proof of the claim.

Denote by $\phi:C^{\prime}\to\Gamma$ the restriction of the morphism
$f^{\prime}:S^{\prime}\to\Gamma$ on $C^{\prime}$. Since
$\mathcal{O}_{Y^{\prime}}(1)$ is induced by the hyperplane section on
$F^{\prime}$, which are in fact the images of the divisors from
$|\Gamma^{\prime}_{0}+E^{\prime}\mathfrak{f}^{\prime}|$, it is sufficient to
see that
$\phi_{\ast}\mathcal{O}_{C^{\prime}}(x^{\prime})=\mathcal{O}_{\Gamma}\oplus\mathcal{O}_{\Gamma}(-E^{\prime})$.
The last follows from [FP05a, Proposition 2.2] because $C^{\prime}\sim
2\Gamma^{\prime}_{0}+(2E^{\prime}+q)\mathfrak{f}^{\prime}$ is bisecant and
$\mathcal{O}_{\Gamma}\oplus\mathcal{O}_{\Gamma}(-E^{\prime})\cong\phi_{\ast}({\mathcal{O}_{S^{\prime}}(\Gamma^{\prime}_{0})}_{|_{C^{\prime}}})=\phi_{\ast}\mathcal{O}_{C^{\prime}}(x^{\prime})$.
This proves the claim. ∎

Using the claim we conclude that
$N_{Y^{\prime}/\mathbb{P}^{R-2}}\otimes{\varphi}^{\prime}_{\ast}\mathcal{O}_{X^{\prime}}(P^{\prime})\cong
N_{Y^{\prime}/\mathbb{P}^{R-2}}\oplus N_{Y^{\prime}/\mathbb{P}^{R-2}}(-1)$. By
the generality of $Y^{\prime}\cong\Gamma$ and [Lop96, Theorem 1.6, Corollary
1.7], we obtain that
$h^{0}(Y^{\prime},N_{Y^{\prime}/\mathbb{P}^{R-2}}(-1))=R-1$, hence

$\displaystyle
h^{0}(X^{\prime},{{\varphi}^{\prime}}^{\ast}N_{Y^{\prime}/\mathbb{P}^{R-2}}\otimes\mathcal{O}_{X^{\prime}}(P^{\prime}))$
$\displaystyle=h^{0}(Y^{\prime},N_{Y^{\prime}/\mathbb{P}^{R-2}})+h^{0}(Y^{\prime},N_{Y^{\prime}/\mathbb{P}^{R-2}}(-1))$
$\displaystyle=R^{2}-g+7\gamma-6\,.$

Using the exact sequence (2) and the non-speciality of
$\mathcal{O}_{X^{\prime}}(1)\otimes\mathcal{O}_{X^{\prime}}(R_{{\varphi}^{\prime}}+2P^{\prime})$
we find

$\displaystyle
h^{0}(X,N_{X^{\prime}/\mathbb{P}^{R-1}})=h^{0}(Y^{\prime},N_{Y^{\prime}/\mathbb{P}^{R-2}})+h^{0}(X^{\prime},{{\varphi}^{\prime}}^{\ast}N_{Y^{\prime}/\mathbb{P}^{R-2}}\otimes\mathcal{O}_{X^{\prime}}(p))=R^{2}+2g-\gamma\,.$

Recall that for the dimension of the family $\mathcal{F}^{\prime}$ we found
$\operatorname{dim}\mathcal{F}^{\prime}=R^{2}+2g-\gamma-1=h^{0}(X,N_{X^{\prime}/\mathbb{P}^{R-1}})-1$.
This implies that either the component $\mathcal{D}^{\prime}_{d-1,g,R-1}$ is
contained properly in a generically smooth component of dimension
$R^{2}+2g-\gamma$ or that $\mathcal{D}^{\prime}_{d-1,g,R-1}$ is non-reduced.
In any cases we have

(3)
$\operatorname{dim}T_{[X^{\prime}]}\mathcal{D}^{\prime}_{d-1,g,R-1}=R^{2}+2g-\gamma\,.$

Step III. To deduce that $\mathcal{D}^{\prime}_{d-1,g,R-1}$ is an irreducible
component of $\mathcal{I}_{d-1,g,R-1}$ it is sufficient to show that
$\mathcal{D}^{\prime}_{d-1,g,R-1}$ is not properly contained in a component of
$\mathcal{I}_{d-1,g,R-1}$. For this assume the opposite, that is, suppose that
there exist an irreducible component
$\mathcal{E}\subset\mathcal{I}_{d-1,g,R-1}$ that contains
$\mathcal{D}^{\prime}_{d-1,g,R-1}$ properly. This implies that
$\operatorname{dim}\mathcal{E}\geq\operatorname{dim}\mathcal{D}^{\prime}_{d-1,g,R-1}+1=R^{2}+2g-\gamma$.
Also, there must exist a flat family $\mathcal{X}$ of curves of genus $g$ and
degree $d-1$ in $\mathbb{P}^{R-1}$ parametrized by an irreducible curve $T$

${\mathcal{X}}$${\mathbb{P}^{R-1}\times T}$${T}$

such that $\mathcal{X}_{t_{0}}$ is a curve from the family
$\mathcal{F}^{\prime}$ for some $t_{0}\in T$ while $\mathcal{X}_{t}$
corresponds to a general point of $\mathcal{E}$ for $t\in T\setminus t_{0}$.
By upper semicontinuity,
$h^{0}(\mathcal{X}_{t_{0}},N_{\mathcal{X}_{t_{0}}/\mathbb{P}^{R-1}})=R^{2}+2g-\gamma=\operatorname{dim}T_{[\mathcal{X}_{t_{0}}]}\mathcal{E}$
and
$\operatorname{dim}\mathcal{E}\geq\operatorname{dim}\mathcal{D}^{\prime}_{d-1,g,R-1}+1=R^{2}+2g-\gamma$
it follows that
$h^{0}(\mathcal{X}_{t},N_{\mathcal{X}_{t}/\mathbb{P}^{R-1}})=R^{2}+2g-\gamma$.
In particular, $\mathcal{E}$ must be generically smooth of dimension
$\operatorname{dim}\mathcal{E}=R^{2}+2g-\gamma$. We will show that is
impossible.

The essence of the arguments that follow is that the gonality of the curves
from the family $\mathcal{F}^{\prime}$ is too large for the family
$\mathcal{F}^{\prime}$ to be contained properly in another family. A general
curve $X^{\prime}$ from $\mathcal{F}^{\prime}$ is double cover of a general
(in moduli sense) curve $Y^{\prime}$ of genus $\gamma$, hence
$\operatorname{gon}(Y^{\prime})=\lfloor\frac{\gamma+3}{2}\rfloor$. The
Castelnuovo-Severi inequality then easily implies that

$\operatorname{gon}(X^{\prime})=2\lfloor\frac{\gamma+3}{2}\rfloor\,.$

We can assume that $\mathcal{X}_{t_{0}}$ is a general curve from
$\mathcal{F}^{\prime}$. Since $\mathcal{X}_{t_{0}}\subset\mathbb{P}^{R-1}$ is
linearly normal, it follows by upper semicontinuity that
$h^{0}(\mathcal{X}_{t},\mathcal{O}_{\mathcal{X}_{t}}(1))=R$. By the Riemann-
Roch theorem we obtain that
$h^{0}(\mathcal{X}_{t},\omega_{\mathcal{X}_{t}}(-1))=\gamma$. Let $B_{t}$ be
the base locus of the linear series $|\omega_{\mathcal{X}_{t}}(-1)|$, and let
$b_{t}:=\deg B_{t}$. Then the linear series
$|\omega_{\mathcal{X}_{t}}(-1)(-B_{t})|$ must be complete of dimension
$\gamma-1$ and degree $4\gamma-3-b_{t}$. We denote
$|\omega_{\mathcal{X}_{t}}(-1)(-B_{t})|=:g^{\gamma-1}_{4\gamma-3-b_{t}}$. Note
that in such a case the Riemann-Roch theorem easily gives that
$|\mathcal{O}_{\mathcal{X}_{t}}(1)(B_{t})|$ is a very ample linear series on
$\mathcal{X}_{t}$, and has dimension $R-1+b_{t}$ and degree $d-1+b_{t}$.

In what follows we will use the variety $\mathcal{W}^{r}_{\delta}(\xi)$ of
line bundles defined on a family of curves $\xi:\mathcal{C}\to N$ parametrized
by a scheme $N$, that is,

$\mathcal{W}^{r}_{\delta}(\xi)=\left\\{(\mathcal{C}_{\nu},L_{\nu})\,\mid\,L_{\nu}\mbox{
is a line bundle of degree }\delta\mbox{ on }\mathcal{C}_{\nu}\mbox{ and
}h^{0}(\mathcal{C}_{\nu},L_{\nu})\geq r+1\right\\}\,.$

For details about them refer to [ACG11, Section 21] or [CIK17]. Remark that to
each irreducible component $\mathcal{W}\subset\mathcal{W}^{r}_{\delta}(\xi)$
corresponds an irreducible component
$\widetilde{\mathcal{W}}\subset\mathcal{W}^{\kappa-\delta+r-1}_{2\kappa-2-\delta}(\xi)$,
where $\mathcal{W}^{\kappa-\delta+r-1}_{2\kappa-2-\delta}(\xi)$ is the dual to
$\mathcal{W}^{r}_{\delta}(\xi)$ variety in the relative Jacobian and $\kappa$
is the genus of the curves from the family $\xi:\mathcal{C}\to N$. Indeed,
$\operatorname{dim}\widetilde{\mathcal{W}}=\operatorname{dim}\mathcal{W}$.

Consider now the component $\mathcal{E}\subset\mathcal{I}_{d-1,g,R-1}$. With a
small abuse of notation, we denote the family of curves parametrized by
$\mathcal{E}$ again by $\mathcal{X}$. Thus, the fibers of the family
$\mathcal{X}\to\mathcal{E}$ are denoted by $\mathcal{X}_{t}$ as we consider
$t\in\mathcal{E}$. Since
$h^{0}(\mathcal{X}_{t},\mathcal{O}_{\mathcal{X}_{t}}(1))=R$, the component
$\mathcal{E}$ must be generically fibered over a component $\mathcal{W}$ of
$\mathcal{W}^{R-1}_{d-1}$ as the fibers consist of the automorphisms of
$\mathbb{P}^{R-1}$. Let $\widetilde{\mathcal{W}}$ be the dual of $\mathcal{W}$
in the relative Jacobian scheme. Remark that
$\widetilde{\mathcal{W}}\subset\mathcal{W}^{\gamma-1}_{4\gamma-3}$. The
general points of $\widetilde{\mathcal{W}}$ are pairs
$(\mathcal{X}_{t},\omega_{\mathcal{X}_{t}}(-1))$. A priori, such
$|\omega_{\mathcal{X}_{t}}(-1))|$ on $\mathcal{X}_{t}$ has base locus $B_{t}$
of degree $b_{t}$. The function that returns the degree of the base locus of a
series in a variety of line bundles on a moving curve is upper semicontinuous.
Hence, there is minimal number, say $b$, such that for every
$(\mathcal{X}_{t},\omega_{\mathcal{X}_{t}}(-1))\in\widetilde{\mathcal{W}}$ the
degree of the base locus of $|\omega_{\mathcal{X}_{t}}(-1)|$ is at least $b$
and for a general $(\mathcal{X}_{t},\omega_{\mathcal{X}_{t}}(-1))$ the degree
is $b_{t}=b$.

Consider the morphism

$\phi_{t}\,:\,\mathcal{X}_{t}\to\mathcal{Z}_{t}\subset\mathbb{P}^{\gamma-1}\,.$

induced by $g^{\gamma-1}_{4\gamma-3-b}$, where
$\mathcal{Z}_{t}:=\phi_{t}(\mathcal{X}_{t})$ and
$g^{\gamma-1}_{4\gamma-3-b}=|\omega_{\mathcal{X}_{t}}(-1)(-B_{t})|$ and
$(\mathcal{X}_{t},\omega_{\mathcal{X}_{t}}(-1))\in\widetilde{\mathcal{W}}$ is
general. If $\mathcal{Z}_{t}$ is not smooth we consider the morphism
$\tilde{\phi}_{t}$ that factors through its normalization
$j\,:\,\tilde{\mathcal{Z}}_{t}\to\mathcal{Z}_{t}$

(4)
${\tilde{\mathcal{Z}}_{t}}$${\mathcal{X}_{t}}$${\mathcal{Z}_{t}}$$\scriptstyle{j}$$\scriptstyle{\tilde{\phi}_{t}}$$\scriptstyle{{\phi}_{t}}$

Denote by $k$ the degree of $\phi_{t}$ (notice that both $\phi_{t}$ and
$\tilde{\phi}_{t}$ are of the same degree). Let $\tau$ be the geometric genus
of $\mathcal{Z}_{t}$, or equivalently, the genus of $\tilde{\mathcal{Z}}_{t}$.
There are two main cases about the morphism $\phi_{t}$ that we will consider
separately:

  * (i)

$k=1$, that is, $\phi_{t}\,:\,\mathcal{X}_{t}\to\mathcal{Z}_{t}$ is a
birational morphism, and

  * (ii)

$k\geq 2$, that is, $\phi_{t}$ is a multi-covering morphism.

We begin with the first case which is easier to deal with. In it we get that
$\mathcal{X}_{t}$ must be birational to its image $\mathcal{Z}_{t}$, hence
$\tau=g$. Recall that by the Castelnuovo genus bound, if $D$ is a smooth
integral curve that admits a birationally very ample linear series
$g^{r^{\prime}}_{d^{\prime}}$ then for its genus $g(D)$ we have

$g(D)\leq\pi(d^{\prime},r^{\prime}):=\binom{m}{2}(r^{\prime}-1)+m\varepsilon\,,$

where $m=\lfloor\frac{d^{\prime}-1}{r^{\prime}-1}\rfloor$ and
$\varepsilon=(d^{\prime}-1)-m(r^{\prime}-1)$, see [ACGH85, p. 116]. We apply
Castelnuovo’s inequality to the curve $\mathcal{X}_{t}$ we get

$g\leq\pi(4\gamma-3-b,\gamma-1)\leq\pi(4\gamma-3,\gamma-1)=6\gamma+4\,,$

which is impossible due to the assumption $g\geq 6\gamma+5$ in the theorem.

We turn now to case (ii). To avoid confusion, we recall that in regards for
the family of $\mathcal{X}\to\mathcal{E}$ we denote by $\mathcal{X}_{t_{0}}$ a
specialization to a curve in $\mathcal{F}^{\prime}$ while $\mathcal{X}_{t}$
stands for a curve corresponding to a general point of $\mathcal{E}$. Recall
that the function returning the gonality of a curve in family is lower
semicontinuous, that is, it doesn’t increase under specialization. As we
remarked already,
$\operatorname{gon}(\mathcal{X}_{t_{0}})=2\lfloor\frac{\gamma+3}{2}\rfloor$,
so

(5)
$\operatorname{gon}(\mathcal{X}_{t})\geq\operatorname{gon}(\mathcal{X}_{t_{0}})\geq
10\quad\mbox{ for }\quad\gamma\geq 7\,.$

The morphism $\phi_{t}$ in (4) induces a linear series
$g^{\gamma-1}_{\frac{4\gamma-3-b}{k}}$ on $\mathcal{Z}_{t}$, or equivalently a
linear series $\tilde{g}^{\gamma-1}_{\frac{4\gamma-3-b}{k}}$ on
$\widetilde{\mathcal{Z}}_{t}$. Obviously, $k\leq 4$ as $k=4$ if and only if
$b=1$ and $\widetilde{\mathcal{Z}}_{t}$ is rational. However, the last would
imply $\operatorname{gon}(\mathcal{X}_{t})\leq 4$, which contradicts with (5).
This leaves $k=2$ or $k=3$ as the only options.

Before proceeding further, let us recall that sub-locus
$\Sigma^{k}_{g,\tau}\subset\mathcal{M}_{g}$ of curves that admit a $k$-sheeted
covering map to a curve of genus $\tau$ has dimension
$\operatorname{dim}\Sigma^{k}_{g,\tau}=2g-2-(2k-3)(\tau-1)$, see for example
[Lan75]. Note also that there is a rational map

$\Phi\,:\,\mathcal{I}_{d-1,g,R-1}\dashrightarrow\mathcal{M}_{g}$

defined on the open subset of points corresponding to smooth integral curves.
In particular, if $\mathcal{E}_{0}\subset\mathcal{E}$ is the open subset of
smooth curves, then $\Phi(\mathcal{E}_{0})\subset\mathcal{M}_{g}$ and

$\operatorname{dim}\Phi(\mathcal{E}_{0})\geq\operatorname{dim}\Phi(\mathcal{D}^{\prime}_{d-1,g,R-1})=2g-\gamma-1\,.$

$\Phi(\mathcal{E}_{0})$ is contained in $\Sigma^{k}_{g,\tau}$ since
$\mathcal{X}_{t}\to\widetilde{\mathcal{Z}}_{t}$ is a $k$-sheeted cover.
Therefore
$\operatorname{dim}\Phi(\mathcal{E}_{0})\leq\operatorname{dim}\Sigma^{k}_{g,\tau}$,
which implies
$2g-2-(2k-3)(\tau-1)\geq\operatorname{dim}\Phi(\mathcal{E}_{0})\geq
2g-\gamma-1$, hence

(6) $\gamma-1\geq(2k-3)(\tau-1)\,.$

Since $\widetilde{\mathcal{Z}}_{t}$ supports
$g^{1}_{\lfloor\frac{\tau+3}{2}\rfloor}$, it follows that
$\operatorname{gon}(\mathcal{X}_{t})\leq\frac{k}{2}(\tau+3)$. Combining the
last with (5) we obtain

(7) $\gamma+2\leq\frac{k}{2}(\tau+3)\ .$

Together (6) and (7) imply that

(8) $\frac{k}{2}(\tau+3)\geq(2k-3)(\tau-1)+3\ ,$

whence either $k=3$ and $\tau\leq 3$ or $k=2$.

Consider $k=3$ and $\tau\leq 3$. Then $\widetilde{\mathcal{Z}}_{t}$ admits a
$g^{1}_{3}$, and hence ${\mathcal{X}_{t}}$ carries a $g^{1}_{9}$ via
$\tilde{\phi}_{t}$ in (4). This is contraction to the equation (5).

It remains to consider the case $k=2$. In this case inequalities (6) and (7)
imply that

$\gamma-1\leq\tau\leq\gamma\,.$

Let $\tilde{H}$ be a divisor on $\widetilde{\mathcal{Z}}_{t}$ corresponding to
a hyperplane section of $\mathcal{Z}_{t}$. Since
$\operatorname{dim}|\tilde{H}|=\tau$, $|\tilde{H}|$ is non-special and hence
$\operatorname{dim}|\tilde{H}+z|=\tau+1$ for any point
$z\in\widetilde{\mathcal{Z}_{t}}$. For $z_{1},z_{2}\in{\mathcal{X}_{t}}$ with
$\phi_{t}(z_{1})\neq\phi_{t}(z_{2})$ and
$\operatorname{supp}(B_{t}\cap(z_{1}+z_{2}))=\emptyset$

$\displaystyle\operatorname{dim}|\omega_{\mathcal{X}_{t}}(-1)+z_{1}+z_{2}|$
$\displaystyle\geq\operatorname{dim}|\omega_{\mathcal{X}_{t}}(-1)(-B_{t}+z_{1}+z_{2})|$
$\displaystyle=\operatorname{dim}|\omega_{\mathcal{X}_{t}}(-1)(-B_{t})|+1$
$\displaystyle=\operatorname{dim}|\omega_{\mathcal{X}_{t}}(-1)|+1\,.$

Hence,

$\operatorname{dim}|\mathcal{O}_{\mathcal{X}_{t}}(-1)(-z_{1}-z_{2})|\geq\operatorname{dim}|\mathcal{O}_{\mathcal{X}_{t}}(-1)|-1\,,$

which contradicts with the very ampleness of
$\mathcal{O}_{\mathcal{X}_{t}}(1)$.

The only possibility that remains to consider is $\tau=\gamma$. Since
$\widetilde{\mathcal{Z}}_{t}$ is of genus $\tau=\gamma$, the existence of
$\tilde{g}^{\gamma-1}_{2\gamma-2-\frac{b-1}{2}}$ on
$\widetilde{\mathcal{Z}}_{t}$ implies $b=1$. This means that
$\Phi(\mathcal{E}_{0})\subset\Sigma^{2}_{g,\gamma}$. Since
$\Phi_{|_{\mathcal{E}_{0}}}\,:\,\mathcal{E}_{0}\to\mathcal{M}_{g}$ is locally
of finite type and

$\operatorname{dim}\Phi^{-1}(\mathcal{X}_{t_{0}})=R^{2}\,,$

it follows by upper semicontinuity that

$\operatorname{dim}\Phi^{-1}(\mathcal{X}_{t})\leq R^{2}\quad\mbox{ for general
}\quad t\in\mathcal{E}\,.$

Therefore
$\operatorname{dim}\mathcal{E}=\operatorname{dim}\mathcal{E}_{0}\leq\operatorname{dim}\Sigma^{2}_{g,\gamma}+R^{2}=R^{2}+2g-\gamma-1$,
which is a contradiction. Alternative argument in this case is to observe that
the line bundle in the pair
$(\mathcal{X}_{t},\mathcal{O}_{\mathcal{X}_{t}}(1))\in\mathcal{W}\subset\mathcal{W}^{R-1}_{d-1}$
is $\mathcal{O}_{\mathcal{X}_{t}}(R_{\tilde{\phi}_{t}}+B_{t})$, where
$R_{\tilde{\phi}_{t}}$ is the ramification divisor of the double covering
$\tilde{\phi}_{t}:\mathcal{X}_{t}\to\widetilde{\mathcal{Z}}_{t}$. Fixing a
double covering completely determines
$\mathcal{O}_{\mathcal{X}_{t}}(R_{\tilde{\phi}_{t}})$, therefore

$\operatorname{dim}\mathcal{E}_{0}\leq\operatorname{dim}\Sigma^{2}_{g,\gamma}+1+\operatorname{dim}\operatorname{Aut}(\mathbb{P}^{R-1})=2g-\gamma+R^{2}-1\,,$

which implies the same numerical contradiction due to
$\operatorname{dim}\mathcal{E}=\operatorname{dim}\mathcal{E}_{0}=2g-\gamma+R^{2}$.
This completes the proof of the theorem.

## References

  * [ACG11] Enrico Arbarello, Maurizio Cornalba, and Pillip A. Griffiths.  Geometry of algebraic curves. Volume II, volume 268 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences].  Springer, Heidelberg, 2011.  With a contribution by Joseph Daniel Harris.
  * [ACGH85] E. Arbarello, M. Cornalba, P. A. Griffiths, and J. Harris.  Geometry of algebraic curves. Vol. I, volume 267 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences].  Springer-Verlag, New York, 1985.
  * [BKMR91] Giorgio Bolondi, Jan O. Kleppe, and Rosa María Miró-Roig.  Maximal rank curves and singular points of the Hilbert scheme.  Compositio Math., 77(3):269–291, 1991.
  * [CCFM09] A. Calabri, C. Ciliberto, F. Flamini, and R. Miranda.  Special scrolls whose base curve has general moduli.  In Interactions of classical and numerical algebraic geometry, volume 496 of Contemp. Math., pages 133–155. Amer. Math. Soc., Providence, RI, 2009.
  * [CIK17] Youngook Choi, Hristo Iliev, and Seonja Kim.  Reducibility of the Hilbert scheme of smooth curves and families of double covers.  Taiwanese J. Math., 21(3):583–600, 2017.
  * [CIK21] Youngook Choi, Hristo Iliev, and Seonja Kim.  Components of the hilbert scheme of smooth projective curves using ruled surfaces.  Manuscr. Math., 164(3-4):395–408, 2021.
  * [CLM96] Ciro Ciliberto, Angelo Felice Lopez, and Rick Miranda.  Some remarks on the obstructedness of cones over curves of low genus.  In Higher-dimensional complex varieties (Trento, 1994), pages 167–182. de Gruyter, Berlin, 1996.
  * [Dan17] Ananyo Dan.  On generically non-reduced components of Hilbert schemes of smooth curves.  Math. Nachr., 290(17-18):2800–2814, 2017.
  * [Ein87] Lawrence Ein.  The irreducibility of the Hilbert scheme of smooth space curves.  In Algebraic geometry, Bowdoin, 1985 (Brunswick, Maine, 1985), volume 46 of Proc. Sympos. Pure Math., pages 83–87. Amer. Math. Soc., Providence, RI, 1987.
  * [Ell87] Philippe Ellia.  D’autres composantes non réduites de ${\rm{Hilb}}{\bf P}^{3}$.  Math. Ann., 277(3):433–446, 1987.
  * [FOV99] Hubert Flenner, Liam O’Carroll, and Wolfgang Vogel.  Joins and intersections. Berlin: Springer, 1999.
  * [FP05a] Luis Fuentes García and Manuel Pedreira.  Canonical geometrically ruled surfaces.  Math. Nachr., 278(3):240–257, 2005.
  * [FP05b] Luis Fuentes García and Manuel Pedreira.  The projective theory of ruled surfaces.  Note di Matematica, 24(1):25–63, 2005.
  * [FS19] Claudio Fontanari and Edoardo Sernesi.  Non-surjective gaussian maps for singular curves on $K3$ surfaces.  Collect. Math., 70(1):107–115, 2019.
  * [GP82] Laurent Gruson and Christian Peskine.  Genre des courbes de l’espace projectif. II.  Ann. Sci. École Norm. Sup. (4), 15(3):401–418, 1982.
  * [Har77] Robin Hartshorne.  Algebraic geometry.  Springer-Verlag, New York-Heidelberg, 1977.  Graduate Texts in Mathematics, No. 52.
  * [Kim01] Seonja Kim.  On the irreducibility of the family of algebraic curves in complex projective space $P^{r}$.  Comm. Algebra, 29(10):4321–4331, 2001.
  * [Kle87] Jan O. Kleppe.  Nonreduced components of the Hilbert scheme of smooth space curves.  In Space curves (Rocca di Papa, 1985), volume 1266 of Lecture Notes in Math., pages 181–207. Springer, Berlin, 1987.
  * [KO15] Jan O. Kleppe and John C. Ottem.  Components of the Hilbert scheme of space curves on low-degree smooth surfaces.  Internat. J. Math., 26(2):1550017, 30, 2015.
  * [Lan75] Herbert Lange.  Moduli spaces of algebraic curves with rational maps.  Math. Proc. Camb. Philos. Soc., 78:283–292, 1975.
  * [Lop96] A. Lopez.  Surjectivity of Gaussian maps on curves in $\mathbb{P}^{r}$ with general moduli.  J. Algebraic Geom., 5 (1996), no. 4, 609–631.
  * [MDP96] Mireille Martin-Deschamps and Daniel Perrin.  Le schéma de Hilbert des courbes gauches localement Cohen-Macaulay n’est (presque) jamais réduit.  Ann. Sci. École Norm. Sup. (4), 29(6):757–785, 1996.
  * [Mum62] D. Mumford.  Further pathologies in algebraic geometry.  Am. J. Math., 84:642–648, 1962.
  * [Nas06] Hirokazu Nasu.  Obstructions to deforming space curves and non-reduced components of the Hilbert scheme.  Publ. Res. Inst. Math. Sci., 42(1):117–141, 2006.
  * [Ser82] Edoardo Sernesi.  Un esempio di curva ostruita in ${\bf P}_{3}$.  Seminario di Variabili Complesse, Bologna 1981, pages 223–231, 1982.
  * [Ser84] E. Sernesi.  On the existence of certain families of curves.  Invent. Math., 75(1):25–57, 1984.
  * [Wal94] Charles H. Walter.  Hyperplane sections of space curves of small genus.  Comm. Algebra, 22(13):5167–5174, 1994.