#  Non-reduced components of the Hilbert scheme of curves
using triple covers

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>
###### Abstract.

In this paper we consider curves on a cone that pass through the vertex and
are also triple covers of the base of the cone, which is a general smooth
curve of genus $\gamma$ and degree $e$ in $\mathbb{P}^{e-\gamma}$. Using the
free resolution of the ideal of such a curve found by Catalisano and
Gimigliano, and a technique concerning deformations of curves introduced by
Ciliberto, we show that the deformations of such curves remain on cones over a
deformation of the base curve. This allows us to prove that for $\gamma\geq 3$
and $e\geq 4\gamma+5$ there exists a non-reduced component $\mathcal{H}$ of
the Hilbert scheme of smooth curves of genus $3e+3\gamma$ and degree $3e+1$ in
$\mathbb{P}^{e-\gamma+1}$. We show that
$\operatorname{dim}T_{[X]}\mathcal{H}=\operatorname{dim}\mathcal{H}+1=(e-\gamma+1)^{2}+7e+5$
for a general point $[X]\in\mathcal{H}$.

###### Key words and phrases:

Hilbert scheme of curves, ruled surfaces, triple coverings, curves on cones.

###### 2020 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 second author was supported by Grant
KP-06-N 62/5 of Bulgarian National Science Fund. The third author was
supported by the National Research Foundation of Korea(NRF) grant funded by
the Korea government(MSIT) (2022R1A2C1005977).

##  1\. Introduction

Let $\mathcal{I}_{d,g,r}$ denote 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 $\mathbb{P}^{r}$. The minimal
possible dimension that a component of $\mathcal{I}_{d,g,r}$ can have is
$\lambda_{d,g,r}:=(r+1)d-(r-3)(g-1)$. Recall that this number is called the
_expected dimension_. Note that
$\lambda_{d,g,r}=h^{0}(C,N_{C/\mathbb{P}^{r}})-h^{1}(C,N_{C/\mathbb{P}^{r}})$,
where $N_{C/\mathbb{P}^{r}}$ is the normal bundle to a curve $C$ represented
by a point of $\mathcal{I}_{d,g,r}$. For the tangent space to a component
$\mathcal{H}$ of $\mathcal{I}_{d,g,r}$ at a point $[C]\in\mathcal{H}$ we have

$\operatorname{dim}\mathcal{H}\leq\operatorname{dim}T_{[C]}\mathcal{H}=h^{0}(C,N_{C/\mathbb{P}^{r}})\,.$

If for a general $[C]\in\mathcal{H}$ we have equality, then the component is
generically smooth. Whenever
$\operatorname{dim}\mathcal{H}<\operatorname{dim}T_{[C]}\mathcal{H}$ at a
general $[C]\in\mathcal{H}$, the component is non-reduced.

After obtaining two series of components of the Hilbert scheme
$\mathcal{I}_{d,g,r}$ of curves in [CIK17, Theorem 4.3, Theorem 4.4], we
realized that the curves parametrized by them are found on cones, which
allowed us to compute $h^{0}(X,N_{C/\mathbb{P}^{r}})$ for a curve $C$
represented by a general point of such component. In this way we were able to
strengthen some of the results proved in [CIK17] and to describe the
corresponding curves in a more geometric fashion. In [CIK21, Theorem A] we
identified a series of generically smooth components of
$\mathcal{I}_{2g-4\gamma+2,g,r}$ for every $\gamma\geq 10$ and $\gamma\leq
r\leq g-3\gamma+2$, which extended [CIK17, Theorem 4.3]. In our paper [CIK22]
we found a series of non-reduced components of
$\mathcal{I}_{2g-4\gamma+1,g,g-3\gamma+1}$ for every $\gamma\geq 7$ and $g\geq
6\gamma+5$. We proved that the corresponding non-reduced components
parametrize curves that lie on cones, pass through the vertex of the
corresponding cone and are double covers of its general hyperplane section,
which is a linearly normal nonspecial curve of genus $\gamma$. We remark that
the non-reduced components from [CIK22] are related to those in [CIK17,
Theorem 4.4].

In the present work we continue our study of smooth curves on cones that pass
through the vertex of a cone and are $m$-covers, $m\geq 3$, of the hyperplane
section of the cone. The main result in the paper concerns the case $m=3$ and
says that under suitable numerical assumptions such families of curves give
rise to non-reduced components of the Hilbert scheme of curves. It is
formulated in the next theorem.

######  Main Theorem.

Assume that $e$ and $\gamma$ are integers such that $e\geq 4\gamma+5$ and
$\gamma\geq 3$. Let

$g:=3\gamma+3e\,,\qquad d:=3e+1\quad\mbox{ and }\quad r:=e-\gamma+1\,.$

Then the Hilbert scheme $\mathcal{I}_{d,g,r}$ possesses a _non-reduced_
component $\mathcal{H}$ such that

  1. (i)

$\operatorname{dim}\mathcal{H}=r^{2}+7e+4$;

  2. (ii)

at a general point $[X]\in\mathcal{H}$ we have
$\operatorname{dim}T_{[X]}\mathcal{H}=\operatorname{dim}\mathcal{H}+1$;

  3. (iii)

a general point $[X]\in\mathcal{H}$ represents a curve $X$ lying on a cone $F$
over a smooth curve $Y$ of genus $\gamma$ and degree $e$ in $\mathbb{P}^{r-1}$
such that

    1. (iii).1.

$X\subset\mathbb{P}^{r}$ is projectively normal and passes through the vertex
$P$ of the cone $F$;

    2. (iii).2.

there is a line $l$ from the ruling of $F$ that is tangent to $X$ at $P$ as
the intersection multiplicity is two;

    3. (iii).3.

the projection from $P$ to the hyperplane in $\mathbb{P}^{r}$ containing the
curve $Y$ induces a morphism $\varphi:X\to Y$ of degree three ;

    4. (iii).4.

the ramification divisor $R_{\varphi}$ is linearly equivalent to the divisor
cut on $X$ by a quadric hypersurface together with $Q_{1}+Q_{2}$, where
$Q_{1}$ and $Q_{2}$ are the remaining two points in which $l$ intersects $X$
besides $P$.

Although our main result fits in the context of [CIK21], [CIK22] and [FS21],
it is independent of them. To obtain it, we develop the approach used in
[CIK22], use the characterization of smooth curves on a cone that pass through
its vertex given in [CG99], and apply similar arguments to those used in
[CLM96] and [Cil87] to deduce that every deformation of a curve from the
family of curves constructed in the theorem yields a curve from the same
family. We remark that the technique used in the proof of our Main Theorem
cannot be by applied in the proof of [CIK22, Theorem B], as we explain the
reasons for this in Remark 10. On the other hand, the possibility for curves
on cones, which are algebraically equivalent to a high degree hypersurface
intersection plus a line, to yield a non-reduced component of the Hilbert
scheme of curves has already been suggested in [CLM96, Remark 4.12]. In this
sense our work was inspired by [CLM96].

The free resolution of the ideal of a smooth curve on a cone passing through
its vertex, obtained by Catalisano and Gimigliano in [CG99], plays an
essential role in the proof of our main result. For this reason we describe
their result in section 2 using a setup that fits the framework of the Main
Theorem. Further in the same section we prove several results about smooth
curves on cones that are $m:1$ covers of the base of the cone and pass through
its vertex. Also, for $m=3$ we prove a technical result, namely Proposition 7,
that plays an important role in the proof of our Main Theorem, which is given
in section 3.

We work over the field $\mathbb{C}$. By _curve_ we understand a smooth
integral projective algebraic curve. Given a line bundle $L$ on a smooth
projective variety $X$, or a divisor $\Delta$ associated to $L$, we denote by
$|L|$ or $|\Delta|$ the complete linear series
$\mathbb{P}\left(H^{0}(X,L)\right)$ on $X$. For a line bundle $L$ and a
divisor $\Delta$ on a variety $X$, we abbreviate, occasionally, the notation
of the line bundle $L\otimes\mathcal{O}_{X}(\Delta)$ to simply $L(\Delta)$. We
use $\sim$ to denote linear equivalence of divisors. Given a finite morphism
$\varphi:X\to Y$ of curves and a divisor $\Delta=\sum n_{i}P_{i}$ on $X$, we
denote by $\varphi(\Delta)$ the divisor $\sum n_{i}\varphi(P_{i})$ on $Y$.
When $X$ is an object of a family, we denote by $[X]$ the corresponding point
of the Hilbert scheme representing the family. For all other definitions and
properties of objects not explicitly introduced in the paper the reader can
refer to [Har77] and [ACGH85].

##  2\.  Preliminary results

In our paper [CIK22] we constructed a series of non-reduced components of the
Hilbert scheme of curves using curves that lie on cones as each curve passes
through the vertex of the corresponding cone. There, we considered only curves
that are double covers of the base of the cone. On the other hand, curves on
cones that are $m:1$ covers of the base, $m\geq 2$, and pass through the
vertex have been studied by Catalisano and Gimigliano in [CG99] with a
different aim. Motivated by an earlier work of Jaffe about smooth curves on a
cone that pass through its vertex, see [Jaf91], Catalisano and Gimigliano
showed in [CG99] that such curves are projectively normal, provided that the
base curve of cone is, and gave a resolution of the ideal of such a curve in
terms of a resolution of the ideal of the base curve. We will formulate below
the main result of [CG99]. For this assume that:

  * $\Gamma$

is a smooth integral curve of genus $\gamma$,

  * $E$

is a divisor of degree $e\geq 2\gamma+1$ on $\Gamma$,

  * $q$

is a point on $\Gamma$,

  * $S$

is the ruled surface
$S=\mathbb{P}(\mathcal{O}_{\Gamma}\oplus\mathcal{O}_{\Gamma}(-E))$,

  * $f$

is the natural projection morphism $f:S\to\Gamma$,

  * $\Gamma_{0}$

is the section of minimal self-intersection of $f:S\to\Gamma$, that is, the
one that corresponds to the exact sequence

$0\to\mathcal{O}_{\Gamma}\to\mathcal{O}_{\Gamma}\oplus\mathcal{O}_{\Gamma}(-E)\to\mathcal{O}_{\Gamma}(-E)\to
0$

with $\Gamma_{0}^{2}=\deg\mathcal{O}_{\Gamma}(-E)=-e$,

  * $\Psi$

is the morphism determined by the linear series $|\Gamma_{0}+E\mathfrak{f}|$
on $S$.

We remark that $\Psi$ is isomorphism away from $\Gamma_{0}$ and contracts
$\Gamma_{0}$ to a point, see [FP05a] for more details. Thus, $\Psi$ maps $S$
into a cone, so we denote by

  * $F$

the image of $S$ under $\Psi$, that is, $F=\Psi(S)$, and

  * $P$

the vertex of $F$, that is, $P=\Psi(\Gamma_{0})$.

Set $r:=\operatorname{dim}|\Gamma_{0}+E\mathfrak{f}|$. Then the embedding
$F\subset\mathbb{P}^{r}$ is complete and the hyperplane sections of $F$ are
the images, under $\Psi$, of the divisors from the linear series
$|\Gamma_{0}+E\mathfrak{f}|$ on $S$. Let

  * $\sigma_{D}$

be a section of $f:S\to\Gamma$ for whose image $\sigma_{D}(\Gamma)=:D$ we have
that

  * $D$

is a smooth curve in the linear series $|\Gamma_{0}+E\mathfrak{f}|$ on $S$,
and let

  * $Y$

be the image of $D$ under $\Psi$, that is, $Y=\Psi(D)$.

The curves $\Gamma$, $D$ and $Y$ are isomorphic to one another since $\Psi$ is
an isomorphism away from $\Gamma_{0}$ and
$D\cdot\Gamma_{0}=(\Gamma_{0}+E\mathfrak{f})\cdot\Gamma_{0}=0$. Also, by
[CIK21, Proposition 1], $r=e-\gamma+1$, and $Y$ is a smooth, linearly normal
curve of genus $\gamma$ and degree $e$ in $\mathbb{P}^{r-1}$. In fact, due to
$e\geq 2\gamma+1$, it follows by [Mum70] that $Y$ is projectively normal.
Thus, we can consider $F$ as a cone in $\mathbb{P}^{r}$ over the projectively
normal curve $Y\subset\mathbb{P}^{r-1}$.

  1. (MS)

_We call the above assortment of assumptions about $\Gamma$, $E$, $q$, $S$,
$f$, $\Gamma_{0}$, $\Psi$, $F$, $P$, $r$, $D$, $\sigma_{D}$ and $Y$, and the
properties we described, the _Main Setup_ , and we abbreviate it as (MS)_.

Catalisano-Gimigliano’s result can now be formulated as follows.

######  Proposition 1.

([CG99, Proposition 2]) Assume the conditions and notations of (MS). Let
$C_{m}\in|m\Gamma_{0}+(mE+q)\mathfrak{f}|$ be general and $X_{m}=\Psi(C_{m})$
be the image of $C_{m}$ on $F$, where $m\geq 2$ is an integer. Then

  1. (a)

$X_{m}$ is a smooth integral projectively normal curve that passes through the
vertex $P$;

  2. (b)

given a free resolution of the ideal sheaf $\mathcal{I}_{Y}$ of $Y$

(1)
$0\to\mathcal{F}_{r-2}\to\mathcal{F}_{r-3}\to\cdots\to\mathcal{F}_{1}\to\mathcal{I}_{Y}\to
0\,,$

with
$\mathcal{F}_{i}=\bigoplus\limits^{\beta_{i}}_{j=1}\mathcal{O}_{\mathbb{P}^{r}}(-\beta_{i,j})$,
$i=1,\ldots,r-2$, the ideal sheaf $\mathcal{I}_{X_{m}}$ of $X_{m}$ has a free
resolution

(2)
$0\to\mathcal{P}_{r-1}\to\mathcal{P}_{r-2}\to\cdots\to\mathcal{P}_{1}\to\mathcal{I}_{X_{m}}\to
0\,,$

where

     * $\mathcal{P}_{1}=\bigoplus\limits^{r-1}_{1}\mathcal{O}_{\mathbb{P}^{r}}(-m-1)\oplus\bigoplus\limits^{\beta_{1}}_{j=1}\mathcal{O}_{\mathbb{P}^{r}}(-\beta_{1,j})$

     * $\mathcal{P}_{k}=\bigoplus\limits^{\binom{r-1}{k}}_{1}\mathcal{O}_{\mathbb{P}^{r}}(-m-k)\oplus\bigoplus\limits^{\beta_{k}}_{j=1}\mathcal{O}_{\mathbb{P}^{r}}(-\beta_{k,j})\oplus\bigoplus\limits^{\beta_{k-1}}_{1}\mathcal{O}_{\mathbb{P}^{r}}(-m-\beta_{k-1,j})$, for $2\leq k\leq r-2$

     * $\mathcal{P}_{r-1}=\mathcal{O}_{\mathbb{P}^{r}}(-m-r+1)\oplus\bigoplus\limits^{\beta_{r-2}}_{1}\mathcal{O}_{\mathbb{P}^{r}}(-m-\beta_{r-2,j})$.

######  Remark 2.

Note that for any point $z\in\Gamma$ the morphism $\Psi$ maps the fiber
$z\mathfrak{f}$ to a line from the ruling of $F$ passing through the point
$\Psi(\sigma_{D}(z))$ on $Y$. Let $l_{q}\subset F$ be the line corresponding
to $q$. As it is pointed out in [CG99, section 1], the curve $X_{m}$, together
with $(e-1)$ lines $L_{1},\ldots,L_{e-1}$ from the ruling of $F$, is cut on
$F$ by a degree $(m+1)$ hypersurface $G_{m+1}\subset\mathbb{P}^{r}$, where
$L_{1},\ldots,L_{e-1}$ are the residual lines on $F$ cut by a hyperplane that
contains the line $l_{q}$. We remark also that the smoothness of a general
$C_{m}\in|m\Gamma_{0}+(mE+q)\mathfrak{f}|$ follows by [Jaf91] and [CG99].

Note that since the curve $C_{m}$ is in linear equivalence class of
$m\Gamma_{0}+(mE+q)\mathfrak{f}$, the adjunction formula gives about its genus
$g$

$\displaystyle 2g-2$
$\displaystyle=(-2\Gamma_{0}+(K_{\Gamma}-E)\mathfrak{f}+m\Gamma_{0}+(mE+q)\mathfrak{f})\cdot(m\Gamma_{0}+(mE+q)\mathfrak{f})$
$\displaystyle=m(m-1)e+2m\gamma-2\,,$

hence $g=\binom{m}{2}e+m\gamma$. Likewise, $(\Gamma_{0}+E\mathfrak{f})\cdot
C_{m}=me+1$, so $X_{m}$ is a smooth curve of degree $d=me+1$ and same genus
$g$. We remark also that if $q_{0}$ is the point in which the fiber
$q\mathfrak{f}$ meets $\Gamma_{0}$, then it follows by [FP05a, Proposition 36]
that the linear series $|m\Gamma_{0}+(mE+q)\mathfrak{f}|$ has a unique base
point at $q_{0}$. This allows us to make the following observation about
$X_{m}$.

######  Proposition 3.

Assume the conditions and notations of (MS). Let $l_{q}$ be as in Remark 2 and
$X_{m}$ be as above. The line $l_{q}$ is tangent to $X_{m}$ at the point $P$
as their intersection multiplicity at $P$ is exactly two.

###### Proof.

The morphism $\Psi:S\to F$ is in fact the resolution of singularity of $F$ at
the vertex $P$. Since $C_{m}$ is the proper transform of $X_{m}$ and
$q\mathfrak{f}$ is the proper transform of $l_{q}$, they wouldn’t meet on
$\Gamma_{0}=\Psi^{-1}(P)$ unless the intersection of $X_{m}$ and $l_{q}$ at
$P$ is of multiplicity at least two. On the other hand
$C_{m}\in|m\Gamma_{0}+(mE+q)\mathfrak{f}|$ is general, hence $q\mathfrak{f}$
meets $C_{m}$ in additional $m-1$ points, all of which are distinct and away
from $\Gamma_{0}$. Since $\Psi$ is an isomorphism away from $\Gamma_{0}$, the
images of those $m-1$ points will be distinct points on $l_{q}$ and away from
$P$. The inner projection with center $P$ to the hyperplane containing $Y$
yields an $m:1$ covering $X_{m}\to Y$, therefore the intersection multiplicity
of $X_{m}$ and $l_{q}$ at $P$ can only be two. ∎

It is convenient to have an explicit notation for the morphism mentioned in
the proof of the lemma, so denote by $\varphi:X_{m}\to Y$ the $m:1$ covering
morphism induced by the inner projection with center $P$ to the hyperplane
containing the curve $Y$. We remark that the image $\varphi(P)$ of the point
$P$ is by definition the point in which the tangent line $l_{q}$ to $X_{m}$ at
$P$ meets the hyperplane, which is the point $\Psi(\sigma_{D}(q))=:Q$.
Consider also the morphism $\phi:C_{m}\to D$ defined as the composition
$\phi:=\sigma_{D}\circ(f_{|_{C_{m}}})$. Note that the morphism $\phi$
coincides with the composition
$(\Psi^{-1})_{|_{Y}}\circ\varphi\circ(\Psi_{|_{C_{m}}})$. Next we will derive
a few facts involving the ramification divisor of $\varphi$ but before that we
summarize, for convenience of the reader, the _additional notations_. We will
refer to them as (AN).

  * (AN)

    * $C_{m}$

is a general curve in the linear series $|m\Gamma_{0}+(mE+q)\mathfrak{f}|$,

    * $q_{0}$.

is the unique base point of $|m\Gamma_{0}+(mE+q)\mathfrak{f}|$; note that
$q_{0}\in\Gamma_{0}$,

    * $X_{m}$

is the image $\Psi(C_{m})\subset F\subset\mathbb{P}^{r}$ of $C_{m}$, which is
smooth according to Remark 2,

    * $\varphi$.

is the $m:1$ covering morphism $\varphi:X_{m}\to Y$ induced by the projection
with center $P$ to the hyperplane in $\mathbb{P}^{r}$ containing $Y$,

    * $\phi$.

is the $m:1$ covering morphism $\phi:C_{m}\to D$ defined as
$\phi:=\sigma_{D}\circ(f_{|_{C_{m}}})$,

    * $Q$.

is the point on $Y$ defined as the image $\Psi(\sigma_{D}(q))$ of
$q\in\Gamma$.

######  Proposition 4.

Assume the conditions and notations (MS) and (AN). Denote by $R_{\varphi}$ be
the ramification divisors of the $\varphi$ and let $B_{\varphi}$ be an “one-
half-the-branch-divisor” of $\varphi$, that is, $B_{\varphi}$ is such that
$2B_{\varphi}\sim\varphi(R_{\varphi})$. Then

  1. (a)

$R_{\varphi}$ is linearly equivalent to the divisor cut on $X_{m}$ by a
hypersurface of degree $(m-1)$ together with the $(m-1)$ points, besides $P$,
in which the line $l_{q}$ meets $X_{m}$ ;

  2. (b)

$\deg R_{\varphi}=(m-1)(me+2)$ ;

  3. (c)

$\varphi^{\ast}\mathcal{O}_{Y}(1)\cong\mathcal{O}_{X_{m}}(1)\otimes\mathcal{O}_{X_{m}}(-P)$
;

  4. (d)

$\mathcal{O}_{Y}(B_{\varphi})\cong\mathcal{O}_{Y}(m(m-1)/2)\otimes\mathcal{O}_{Y}((m-1)Q)$
.

###### Proof.

Since $C_{m}$ and $X_{m}$ are isomorphic, we can transform some of the claims
about $R_{\varphi}$ into claims about the ramification divisor $R_{\phi}$ of
the morphism $\phi$, which are easier to prove.

  1. (a)

For $R_{\phi}$ we have

$\displaystyle R_{\phi}$ $\displaystyle\sim K_{C_{m}}-\phi^{\ast}K_{D}$
$\displaystyle\sim(-2\Gamma_{0}+(K_{\Gamma}-E)\mathfrak{f}+m\Gamma_{0}+(mE+q)\mathfrak{f})_{|_{C_{m}}}-\phi^{\ast}K_{D}$
$\displaystyle\sim((m-2)\Gamma_{0}+((m-1)E+q)\mathfrak{f})_{|_{C_{m}}}+K_{\Gamma}\mathfrak{f}_{|_{C_{m}}}-\phi^{\ast}K_{D}\,.$

The divisor $K_{D}$ is the restriction of
$K_{S}+D\sim-2\Gamma_{0}+(K_{\Gamma}-E)\mathfrak{f}+\Gamma_{0}+E\mathfrak{f}\sim-\Gamma_{0}+K_{\Gamma}\mathfrak{f}$
on the curve $D$. However, $D$ doesn’t meet $\Gamma_{0}$, therefore
$\phi^{\ast}K_{D}\sim\phi^{\ast}((-\Gamma_{0}+K_{\Gamma}\mathfrak{f})_{|_{D}})=\phi^{\ast}(K_{\Gamma}\mathfrak{f}_{|_{D}})\sim
K_{\Gamma}\mathfrak{f}_{|_{C_{m}}}$. Therefore

$R_{\phi}\sim((m-2)\Gamma_{0}+((m-1)E+q)\mathfrak{f})_{|_{C_{m}}}\,.$

By the commutativity of the diagram

(3)
${C_{m}}$${X_{m}}$${D}$${Y}$$\scriptstyle{\Psi_{|_{C_{m}}}}$$\scriptstyle{\cong}$$\scriptstyle{\phi}$$\scriptstyle{\varphi}$$\scriptstyle{\Psi_{|_{D}}}$$\scriptstyle{\cong}$

we have that the restriction of $\Psi$ on $C_{m}$ takes a divisor that is
linearly equivalent to the ramification divisor of $\phi$ into a divisor that
is linearly equivalent to the ramification divisor of $\varphi$. Consider

$R_{\phi}\sim((m-2)\Gamma_{0}+((m-1)E+q)\mathfrak{f})_{|_{C_{m}}}\sim((m-1)(\Gamma_{0}+E\mathfrak{f})+(q\mathfrak{f}-\Gamma_{0}))_{|_{C_{m}}}\,.$

Since $\Gamma_{0}$ and $C_{m}$ meet exactly at the point $q_{0}$ in which the
fiber $q\mathfrak{f}$ meets $\Gamma_{0}$, it follows that
$(q\mathfrak{f}-\Gamma_{0}))_{|_{C_{m}}}$ is an effective divisor on $C_{m}$
that consists of $m-1$ points, say $q_{1},\ldots,q_{m-1}$ on $C_{m}$, in which
$q\mathfrak{f}$ intersects $C_{m}$ besides $q_{0}$. Hence,

$R_{\phi}\sim((m-1)(\Gamma_{0}+E\mathfrak{f}))_{|_{C_{m}}}+q_{1}+\cdots+q_{m-1}\,.$

The morphism $\Psi:S\to\mathbb{P}^{r}$ is defined by the linear series
$|\Gamma_{0}+E\mathfrak{f}|$ on $S$, so $\Psi$ maps the restriction
$((m-1)(\Gamma_{0}+E\mathfrak{f}))_{|_{C_{m}}}$ to the divisor on $X_{m}$ cut
by a hypersurface of degree $m-1$. Also, $\Psi$ maps the fiber $q\mathfrak{f}$
into the line $l_{q}$. The images of the points $q_{1},\ldots,q_{m-1}$ under
$\Psi$ will be the $m-1$ points in which $l_{q}$ meets $X_{m}$ besides $P$.
Therefore, $R_{\varphi}$ is linearly equivalent to the divisor cut on $X_{m}$
by a hypersurface of degree $(m-1)$ together with the images of the points
$q_{1},\ldots,q_{m-1}$, which lie on $l_{q}$ as claimed.

  2. (b)

Since $\deg
X_{m}=(\Gamma_{0}+E\mathfrak{f})\cdot(m\Gamma_{0}+(mE+q)\mathfrak{f})=me+1$,
it follows by (a) that

$\displaystyle\deg R_{\varphi}$ $\displaystyle=(m-1)\deg
X_{m}+(m-1)=(m-1)(me+2)\,.$

  3. (c)

The claim and its proof are contained in the proof of [CIK22, Proposition 2].

  4. (d)

To verify the last claim we show for the branch divisor $\phi(R_{\phi})$ of
$\phi:C_{m}\to D$ that

$\phi(R_{\phi})\sim((m(m-1)E+2(m-1)q)\mathfrak{f})_{|_{D}}\,.$

Recall first that the map
$\phi:\operatorname{Div}(C_{m})\to\operatorname{Div}(D)$ is linear in the
sense that $\phi(\sum n_{P_{j}}P_{j})=\sum n_{P_{j}}\phi(P_{j})$. Note also
that according to [Har77, Ex. IV.2.6, p. 306], $\phi(\sum n_{P_{j}}P_{j})$
depends only on the linear equivalence class of $\sum n_{P_{j}}P_{j}$. As we
claim just linear equivalence, it is sufficient to verify that

    1. (1)

$\phi\left((m-2)(\Gamma_{0}+E\mathfrak{f})_{|_{C_{m}}}\right)\sim((m(m-2)E+(m-2)q)\mathfrak{f})_{|_{D}}$,
and

    2. (2)

$\phi\left((E+q)\mathfrak{f})_{|_{C_{m}}}\right)\sim((m(E+q))\mathfrak{f})_{|_{D}}$.

The first claim follows easily from the fact that $\Gamma_{0}$ and $C_{m}$
intersect exactly at $q_{0}$, and $\phi(q_{0})=q\mathfrak{f}_{|_{D}}$ and
$\phi:C_{m}\to D$ being an $m:1$ covering. Similarly with the second claim.
This implies about the branch divisor on $D$ that

$\displaystyle\phi(R_{\phi})$
$\displaystyle\sim\phi((m-2)(\Gamma_{0}+E\mathfrak{f})_{|_{C_{m}}})+\phi\left((E+q)\mathfrak{f})_{|_{C_{m}}}\right)$
$\displaystyle\sim((m(m-2)E+(m-2)q)\mathfrak{f})_{|_{D}}+((m(E+q))\mathfrak{f})_{|_{D}}$
$\displaystyle\sim((m(m-1)E+2(m-1)q)\mathfrak{f})_{|_{D}}\,.$

By the commutativity of diagram (3) we have that

$\varphi(R_{\varphi})\sim\Psi_{|_{D}}(\phi(R_{\phi}))\sim\Psi_{|_{D}}(\phi(((m(m-1)E+2(m-2)q)\mathfrak{f})_{|_{D}}))\,.$

Recall that $\Gamma_{0}$ and $D$ do not intersect and
$E\mathfrak{f}_{|_{D}}\equiv(\Gamma_{0}+E\mathfrak{f})_{|_{D}}$. Since the
divisors from $|\Gamma_{0}+E\mathfrak{f}|$ are mapped by $\Psi$ into
hyperplane sections of $F$, it follows that the branch divisor
$\varphi(R_{\varphi})$ is linearly equivalent to a divisor on $D$ cut by a
hypersurface of degree $m(m-1)$ together with the with the divisor $2(m-1)Q$,
where $Q$ is the point in which the line $l_{q}$ meets $Y$. Therefore, for the
“one-half-the-branch-divisor” $B_{\varphi}$ on $Y$ we obtain that

$\mathcal{O}_{Y}(B_{\varphi})\cong\mathcal{O}_{Y}\left(\frac{m(m-1)}{2}\right)\otimes\mathcal{O}_{Y}((m-1)Q)\,.$

∎

We remark explicitly the following fact which is derived easily from the proof
of Proposition 4(d).

######  Remark 5.

If $\nu:C_{m}\to\Gamma$ is the morphism defined as the restriction of $f$ on
$C_{m}$ and $B_{\nu}$ is the “one-half-the-branch-divisor” of $\nu$, that is,
$2B_{\nu}\sim\nu(R_{\nu})$, where $R_{\nu}$ is the ramification divisor of
$\nu$, then $B_{\nu}$ is linearly equivalent to the divisor
$\frac{m(m-1)}{2}E+(m-1)q$ on $\Gamma$.

The proposition that follows will be used in the proof of the Main Theorem to
identify the curves of given degree and genus that lie a cone in terms of the
linear equivalence class of a specific divisor on the desingularization of the
cone.

######  Proposition 6.

Suppose that $F\subset\mathbb{P}^{r}$ is a cone over a smooth integral
linearly normal curve $Y$ of genus $\gamma$ and degree $e\geq 2\gamma+1$ in
$\mathbb{P}^{r-1}$. Let $S$ be the ruled surface defined as the blow-up of $F$
at its vertex, and let $f:S\to Y$ be the natural surjective morphism with a
section $Y_{0}$ of minimal self-intersection. If $X$ is a smooth integral
curve of degree $d=me+1$ and genus $g=\binom{m}{2}e+m\gamma$ on $F$, then its
proper transform $C$ on $S$ is linearly equivalent to
$mY_{0}+(mE+q)\mathfrak{f}$, where $E$ is an effective divisor of degree $e$
on $Y$ such that $S\cong\mathbb{P}(\mathcal{O}_{Y}\oplus\mathcal{O}_{Y}(-E))$
and $q$ is a point on $Y$.

###### Proof.

Since $S$ is the blow-up of $F$ at its vertex, it must be a decomposable ruled
surface over $Y$. Since $Y_{0}$ is the section of minimal self-intersection of
$f:S\to Y$, we must have that $\deg E=-Y^{2}_{0}=\deg Y=e$. The Picard group
of $S$ is generated by $Y_{0}$ and the pull-backs via $f^{\ast}$ of the
divisors on $Y$. Hence, $C\sim aY_{0}+B\mathfrak{f}$ for a divisor $B$ on $Y$.
For the degree of $X$ we have

$me+1=\deg X=(Y_{0}+E\mathfrak{f})\cdot(aY_{0}+B\mathfrak{f})=-ae+ae+\deg
B\,,$

so $\deg B=me+1$. Applying the adjunction theorem for $C$ we get

$\displaystyle 2g-2$ $\displaystyle=(K_{C}+C)\cdot C$
$\displaystyle=(-2Y_{0}+(K_{Y}-E)\mathfrak{f}+aY_{0}+B\mathfrak{f})\cdot(aY_{0}+B\mathfrak{f})$
$\displaystyle=((a-2)Y_{0}+(K_{Y}-E+B)\mathfrak{f})\cdot(aY_{0}+B\mathfrak{f})$
$\displaystyle=a(a-2)(-e)+(me+1)(a-2)+a(2\gamma-2-e+me+1)$
$\displaystyle=-ea^{2}+2ae+(me+1)a-2me-2+(2\gamma-1+(m-1)e)a\,.$

Since $2g-2=m(m-1)e+2m\gamma-2$, we obtain

(4) $ea^{2}-((2m+1)e+2\gamma)a+m(m+1)e+2m\gamma=0\,.$

Solving (4) for $a$ we obtain solutions $a=m$ and $a=m+1+\frac{2\gamma}{e}$.
Since $e\geq 2\gamma+1$, the second number is not an integer, so $a=m$ is the
only solution.

It remains to prove the claim about $B$, that is, $B\sim mE+q$ for some point
$q\in Y$. An argument similar to that in [Har77, Prop. V.2.6, p.371] shows
that $j_{\ast}\mathcal{O}_{Y_{0}}(Y_{0})\cong\mathcal{O}_{Y}(-E)$, where $j$
is the isomorphism $j:Y_{0}\to Y$. Namely, consider the exact sequence

$0\to\mathcal{O}_{S}\to\mathcal{O}_{S}(Y_{0})\to\mathcal{O}_{Y_{0}}\otimes\mathcal{O}_{S}(Y_{0})\to
0\,,$

and push it down to $Y$. By Grauert’s theorem we have

$0\to f_{\ast}\mathcal{O}_{S}\to f_{\ast}\mathcal{O}_{S}(Y_{0})\to
j_{\ast}(\mathcal{O}_{Y_{0}}(Y_{0}))\to 0\,.$

Since $f_{\ast}\mathcal{O}_{S}\equiv\mathcal{O}_{Y}$ and
$f_{\ast}\mathcal{O}_{S}(Y_{0})\cong\mathcal{O}_{Y}\oplus\mathcal{O}_{Y}(-E)$,
we deduce that $j_{\ast}\mathcal{O}_{Y_{0}}(Y_{0})\cong\mathcal{O}_{Y}(-E)$.
Further, $C$ is a smooth curve on $S$ and $C\cdot
Y_{0}=(mY_{0}+B\mathfrak{f})\cdot Y_{0}=1$, so $C$ intersects $Y_{0}$ in a
single point, say $z=C\cap Y_{0}$. Since $C\sim mY_{0}+B\mathfrak{f}$, the
restrictions $C_{|_{Y_{0}}}$ and $(mY_{0}+B\mathfrak{f})_{|_{Y_{0}}}$ must be
linearly equivalent too. Hence,

$z\sim(mY_{0}+B\mathfrak{f})_{|_{Y_{0}}}\,,$

or equivalently, $j(z)\sim-mE+B$ on $Y_{0}$. Taking $q:=j(z)$, we obtain
$B\sim mE+q$. ∎

In the proof of the main theorem in section 3 we will use
$\varphi_{\ast}\mathcal{O}_{X_{m}}$ and
$\varphi_{\ast}(\mathcal{O}_{X_{m}}(P))$, or equivalently
$\phi_{\ast}\mathcal{O}_{C_{m}}$ and
$\phi_{\ast}(\mathcal{O}_{C_{m}}(\Gamma_{0}))$. The statement giving explicit
expressions of those bundles develops an idea encountered in [FP05b,
Proposition 2.2]. However, due to obvious reasons, we give a formulation and a
proof only in the case $m=3$, which is sufficient for our purposes.

######  Proposition 7.

Assume the conditions and notations (MS) and (AN). Fix $m=3$ and denote
$C_{3}=:C$ and $X_{3}=:X$. Then

  1. (a)

$\varphi_{\ast}(\mathcal{O}_{X}(P))\cong\mathcal{O}_{Y}\oplus\mathcal{O}_{Y}(-1)\oplus(\mathcal{O}_{Y}(-2)\otimes\mathcal{O}_{Y}(-Q))$

  2. (b)

$\varphi_{\ast}\mathcal{O}_{X}\cong\mathcal{O}_{Y}\oplus(\mathcal{O}_{Y}(-1)\otimes\mathcal{O}_{Y}(-Q))\oplus(\mathcal{O}_{Y}(-2)\otimes\mathcal{O}_{Y}(-Q))$

###### Proof.

The corresponding statements for $\phi:C\to D$ appear as

  1. (a’)

$\phi_{\ast}(\mathcal{O}_{C}(\Gamma_{0}))\cong\mathcal{O}_{D}\oplus\mathcal{O}_{D}(-E\mathfrak{f})\oplus\mathcal{O}_{D}(-(2E+q)\mathfrak{f})$,

  2. (b’)

$\phi_{\ast}\mathcal{O}_{C}\cong\mathcal{O}_{D}\oplus\mathcal{O}_{D}(-(E+q)\mathfrak{f})\oplus\mathcal{O}_{D}(-(2E+q)\mathfrak{f})$.

If we denote by $\nu$ the morphism $f_{|_{C}}:C\to\Gamma$, or equivalently, if
$\iota$ is the embedding $\iota:C\hookrightarrow S$ and $\nu$ is the
composition $f\circ\iota$, the claims translate into

  1. (a”)

$\nu_{\ast}(\mathcal{O}_{C}(\Gamma_{0}))\cong\mathcal{O}_{\Gamma}\oplus\mathcal{O}_{\Gamma}(-E)\oplus\mathcal{O}_{\Gamma}(-2E-q)$,

  2. (b”)

$\nu_{\ast}\mathcal{O}_{C}\cong\mathcal{O}_{\Gamma}\oplus\mathcal{O}_{\Gamma}(-E-q)\oplus\mathcal{O}_{\Gamma}(-2E-q)$.

It is sufficient to prove claims (a”) and (b”), which we will do next. We
remark that claim (b”) has been proved by Miranda for varieties of arbitrary
dimension, see [Mir85, Proposition 8.1, p.1150]. For convenience of the reader
we give a proof of it as well as it is easy to do in our context.

Since $C\in|3\Gamma_{0}+(3E+q)\mathfrak{f}|$, there is an exact sequence

$0\to\mathcal{O}_{S}(-2\Gamma_{0}-(3E+q)\mathfrak{f})\to\mathcal{O}_{S}(\Gamma_{0})\to\iota_{\ast}\mathcal{O}_{C}(\Gamma_{0})\equiv\iota_{\ast}\mathcal{O}_{C}(q_{0})\to
0\,.$

Pushing it down to $\Gamma$ via $f_{\ast}$, we get the exact sequence

(5)
${0}$${f_{\ast}\mathcal{O}_{S}(-2\Gamma_{0}-(3E+q)\mathfrak{f})}$${f_{\ast}\mathcal{O}_{S}(\Gamma_{0})}$${f_{\ast}\iota_{\ast}\mathcal{O}_{C}(q_{0})}$${R^{1}f_{\ast}\mathcal{O}_{S}(-2\Gamma_{0}-(3E+q)\mathfrak{f})}$${R^{1}f_{\ast}\mathcal{O}_{S}(\Gamma_{0})}$${R^{1}f_{\ast}\iota_{\ast}\mathcal{O}_{C}(q_{0})}$

For every point $z\in\Gamma$ we have that $z\mathfrak{f}\cong\mathbb{P}^{1}$
and
$\deg(-2\Gamma_{0}-(3E+q)\mathfrak{f})_{|_{z\mathfrak{f}}})=\deg(-2\Gamma_{0}\cdot\mathfrak{f})=-2$,
hence

$h^{i}(z\mathfrak{f},\mathcal{O}_{S}(-2\Gamma_{0}-(3E+q)\mathfrak{f})_{|_{z\mathfrak{f}}})=h^{i}(\mathbb{P}^{1},\mathcal{O}_{\mathbb{P}^{1}}(-2))=\begin{cases}0&\mbox{
if }i=0\\\ 1&\mbox{ if }i=1\end{cases}$

By Grauert’s theorem, see [Har77, Theorem III.12.9], it follows that
$f_{\ast}\mathcal{O}_{S}(-2\Gamma_{0}-(3E+q)\mathfrak{f})$ vanishes, while
$R^{1}f_{\ast}\mathcal{O}_{S}(-2\Gamma_{0}-(3E+q)\mathfrak{f})$ must be a
locally free sheaf of rank one, that is, a line bundle on $\Gamma$. From the
definition of $S$ we have
$f_{\ast}\mathcal{O}_{S}(\Gamma_{0})\cong\mathcal{O}_{\Gamma}\oplus\mathcal{O}_{\Gamma}(-E)$,
and since
$h^{1}(z\mathfrak{f},\mathcal{O}_{S}(\Gamma_{0})_{|_{z\mathfrak{f}}})=h^{1}(\mathbb{P}^{1},\mathcal{O}_{\mathbb{P}^{1}}(1))=0$,
the Grauert’s theorem implies that (5) reduces to

(6)
$0\to\mathcal{O}_{\Gamma}\oplus\mathcal{O}_{\Gamma}(-E)\to\nu_{\ast}\mathcal{O}_{C}(q_{0})\to
R^{1}f_{\ast}\mathcal{O}_{S}(-2\Gamma_{0}-(3E+q)\mathfrak{f})\to 0\,.$

Since $\nu:C\to\Gamma$ is a triple covering morphism, the push-forward
$\nu_{\ast}\mathcal{O}_{C}$ must split as

$\nu_{\ast}\mathcal{O}_{C}=\mathcal{O}_{\Gamma}\oplus\mathcal{E}^{\vee}\,,$

where $\mathcal{E}$ is a vector bundle of rank two on $\Gamma$ for which its
dual bundle $\mathcal{E}^{\vee}$ is the Tschirnhausen module of $\nu$. By
[Har77, Ex. IV.2.6(d), p.306] and Remark 5, it follows that
$\det(\nu_{\ast}\mathcal{O}_{C})=\det\mathcal{E}^{\vee}=\mathcal{O}_{\Gamma}(-3E-2q)$.
Since ${\Gamma_{0}}$ and $C$ meet exactly at the point $q_{0}$ which is mapped
by $\nu$ into $q$ on $\Gamma$, we obtain by [Har77, Ex. IV.2.6(a), p.306] that

$\det(\nu_{\ast}\mathcal{O}_{C}(q_{0}))\cong\det(\nu_{\ast}\mathcal{O}_{C})\otimes\mathcal{O}_{\Gamma}(q)\cong\mathcal{O}_{Y}(-3E-q)\,.$

Therefore

$\displaystyle R^{1}f_{\ast}\mathcal{O}_{S}(-2\Gamma_{0}-(3E+q)\mathfrak{f})$
$\displaystyle\cong\det(\nu_{\ast}\mathcal{O}_{C}(q_{0}))\otimes(\det(\mathcal{O}_{\Gamma}\oplus\mathcal{O}_{\Gamma}(-E))^{-1}$
$\displaystyle\cong\mathcal{O}_{Y}(-3E-q)\otimes\mathcal{O}_{Y}(E)$
$\displaystyle\cong\mathcal{O}_{Y}(-2E-q)\,.$

Since

$\displaystyle\operatorname{Ext}^{1}(R^{1}f_{\ast}\mathcal{O}_{S}(-2\Gamma_{0}-(3E+q)\mathfrak{f}),\mathcal{O}_{\Gamma}\oplus\mathcal{O}_{\Gamma}(-E))$
$\displaystyle=\operatorname{Ext}^{1}(\mathcal{O}_{Y}(-2E-q),\mathcal{O}_{\Gamma}\oplus\mathcal{O}_{\Gamma}(-E))$
$\displaystyle=H^{1}(\Gamma,\mathcal{O}_{\Gamma}(2E+q)\oplus\mathcal{O}_{\Gamma}(E+q))$
$\displaystyle=0\,,$

the exact sequence (6) splits and we obtain
$\nu_{\ast}\mathcal{O}_{C}(q_{0})\cong\mathcal{O}_{\Gamma}\oplus\mathcal{O}_{\Gamma}(-E)\oplus\mathcal{O}_{Y}(-2E-q)$.
This completes the proof of (a).

In order to prove (b), we recall that $\nu:C\to\Gamma$ is a triple covering
for which the rank two vector bundle $\mathcal{E}$ on $\Gamma$ is its
Tschirnhausen module, that is,
$\nu_{\ast}\mathcal{O}_{C}\cong\mathcal{O}_{\Gamma}\oplus\mathcal{E}^{\vee}$.
Since the Tschirnhausen module of the triple cover $\nu:C\to\Gamma$ is
determined uniquely up to a twist by a line bundle on $\Gamma$ and here
$\iota:C\hookrightarrow
S=\mathbb{P}(\mathcal{O}_{\Gamma}\oplus\mathcal{O}_{\Gamma}(-E))$, it follows
by [CE96, Theorem 1.3, p.439] that

$\nu_{\ast}\mathcal{O}_{C}\cong\mathcal{O}_{\Gamma}\oplus((\mathcal{O}_{\Gamma}\oplus\mathcal{O}_{\Gamma}(-E))\otimes\mathcal{L})\,,$

for some line bundle $\mathcal{L}$ on $\Gamma$. As it was seen in (a),
$\det(\nu_{\ast}\mathcal{O}_{C})\cong\mathcal{O}_{\Gamma}(-3E-2q)$, from where
we obtain $\mathcal{L}\cong\mathcal{O}_{\Gamma}(-E-q)$. Therefore

$\nu_{\ast}\mathcal{O}_{C}\cong\mathcal{O}_{\Gamma}\oplus((\mathcal{O}_{\Gamma}\oplus\mathcal{O}_{\Gamma}(-E))\otimes\mathcal{L})\cong\mathcal{O}_{\Gamma}\oplus\mathcal{O}_{\Gamma}(-E-q)\oplus\mathcal{O}_{\Gamma}(-2E-q)\,.$

∎

Finally, we recall one more result that will be used in the proof of the Main
Theorem in section 3.

######  Proposition 8.

([CIK22, Proposition 2]) Let $X$ be a non-degenerate smooth integral curve in
$\mathbb{P}^{r}$, where $r\geq 3$. Let $H$ be a hyperplane in $\mathbb{P}^{r}$
and $P$ be a point on $X$. Suppose that the inner projection $\varphi\,:\,X\to
H\cong\mathbb{P}^{r-1}$ with center $P$ maps $X$ to a non-degenerate smooth
integral curve $Y$ in $H$. Denote by $R_{\varphi}$ the ramification divisor of
$\varphi$. Then

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

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

##  3\.  Proof of the main theorem

Recall the basic numerical assumptions in the theorem: $\gamma\geq 3$ and
$e\geq 4\gamma+5$. Throughout this section we also fix

$g:=3e+3\gamma\,,\quad d:=3e+1=g-3\gamma+1\quad\mbox{ and }\quad
r:=e-\gamma+1=\frac{g}{3}-2\gamma+1\,.$

The technique used in the proof is derived from [CIK22], [CLM96] and [Cil87].
The proof itself proceeds in three main steps:

  * Step I.

We construct a family $\mathcal{F}$ of curves satisfying the condition of the
theorem, then we consider the closure $\mathcal{H}$ of the subset of
$\mathcal{I}_{d,g,r}$ parametrizing the family $\mathcal{F}$ and show that

$\operatorname{dim}\mathcal{H}=r^{2}+7e+4\,.$

  * Step II.

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

$\operatorname{dim}T_{[X]}\mathcal{H}=h^{0}(X,N_{X/\mathbb{P}^{r}})=r^{2}+7e+5=\operatorname{dim}\mathcal{H}+1\,.$

  * Step III.

We show that $\mathcal{H}$ forms an irreducible component of
$\mathcal{I}_{d,g,r}$.

Step I. Construction of the family.

Let $\Gamma\in\mathcal{M}_{\gamma}$ be a general curve of genus $\gamma$ and
$E$ be a general divisor of degree $e\geq 4\gamma+5$ on $\Gamma$. Let
$q\in\Gamma$. Consider the ruled surface
$S:=\mathbb{P}(\mathcal{O}_{\Gamma}\oplus\mathcal{O}_{\Gamma}(-E))$ with
natural projection $f:S\to\Gamma$. Denote by $\Gamma_{0}$ the section of
minimal self-intersection on $S$, that is, $\Gamma^{2}_{0}=-e$. As it was
mentioned in section 2,
$\operatorname{Pic}(S)\cong\mathbb{Z}[\Gamma_{0}]\oplus{f}^{\ast}(\operatorname{Pic}(\Gamma))$.
Just as there, for a divisor $\Delta\in\operatorname{Div}(\Gamma)$ we denote
by $\Delta\mathfrak{f}$ the divisor ${f}^{\ast}(\Delta)$ on $S$. Consider the
morphism $\Psi:=\Psi_{|\Gamma_{0}+E\mathfrak{f}|}:S\to\mathbb{P}^{r}$
determined by the linear series $\mathcal{O}_{S}(\Gamma_{0}+E\mathfrak{f})$ on
$S$. Define $\mathcal{F}$ as the family of curves that are images of the
divisors from the linear series $|3{\Gamma}_{0}+(3E+q){\mathfrak{f}}|$ on $S$
under the morphism $\Psi$, by varying $\Gamma$ in $\mathcal{M}_{\gamma}$,
running $E$ through the set of all effective divisors of degree $e$ on
$\Gamma$ and $q\in\Gamma$. For the dimension of $\mathcal{F}$ we have

$\operatorname{dim}\mathcal{F}=$

  * $+$

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

  * $+$

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

  * $+$

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

  * $+$

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

  * $-$

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

  * $+$

$6e-3\gamma+6=\operatorname{dim}|3{\Gamma}_{0}+(3E+q){\mathfrak{f}}|$ : number
of parameters to choose a curve in the linear equivalence class of
$3{\Gamma}_{0}+(3E+q){\mathfrak{f}}$ on $S$.

Define $\mathcal{H}$ as the closure in $\mathcal{I}_{d,g,r}$ of the set
parametrizing $\mathcal{F}$. Accounting the above numbers we get

$\operatorname{dim}\mathcal{H}=\operatorname{dim}\mathcal{F}=r^{2}+7e+4\,.$

This completes Step I of the proof.

Step II. Computation of the tangent space to $\mathcal{H}$.

Let $X\in\mathcal{F}$ be a general curve from the family, that is, $X$ is the
image $\Psi(C)$ of a general $C\in|3\Gamma_{0}+(3E+q)\mathfrak{f}|$ on $S$,
the base curve $\Gamma\in\mathcal{M}_{\gamma}$ is general, and
$E\in\operatorname{Div}^{e}(\Gamma)$ and $q\in\Gamma$ are also general. Also,
$X$ lies on the cone $F:=\Psi(S)$ over a curve $Y\subset\mathbb{P}^{r-1}$ that
is the image $Y:=\Psi(D)$ of a general $D\in|\Gamma_{0}+E\mathfrak{f}|$. Let
$l_{q}$ be the line from the ruling of $F$ that is the image of
$q\mathfrak{f}$ and $Q=l_{q}\cap Y$. Denote by $\varphi:X\to Y$ the projection
with center $P$ of $X$ to the hyperplane containing $Y$. It is a $3:1$
covering morphism. By Proposition 4 its ramification divisor $R_{\varphi}$ is
linearly equivalent to the divisor on $X$ cut by a quadric hypersurface and
the two points, say $Q_{1}$ and $Q_{2}$, besides $P$, in which the line
$l_{q}$ meets $X$. Applying Proposition 8, we obtain the short exact sequence

(7) $0\to\mathcal{O}_{X}(3)\otimes\mathcal{O}_{X}(Q_{1}+Q_{2}+2P)\to
N_{X/\mathbb{P}^{r}}\to\varphi^{\ast}N_{Y/\mathbb{P}^{r-1}}\otimes\mathcal{O}_{X}(P)\to
0\,,$

in which $N_{X/\mathbb{P}^{r}}$ is the normal bundle of $X$ in
$\mathbb{P}^{r}$ and $N_{Y/\mathbb{P}^{r-}}$ is the normal bundle of $Y$ in
$\mathbb{P}^{r-1}$. Due to $e\geq 2\gamma+1$, the Hilbert scheme
$\mathcal{I}_{e,\gamma,e-\gamma}$ is irreducible and generically smooth of the
expected dimension
$\operatorname{dim}\mathcal{I}_{e,\gamma,e-\gamma}=\lambda_{e,\gamma,r-1}=er-(r-4)(\gamma-1)$.
Since $\Gamma\in\mathcal{M}_{\gamma}$ is general and $Y$ is isomorphic to
$\Gamma$ and $\deg Y=e$, it follows that

$h^{0}(Y,N_{Y/\mathbb{P}^{r-1}})=er-(r-4)(\gamma-1)\,.$

For the degree of the line bundle
$\mathcal{O}_{X}(3)\otimes\mathcal{O}_{X}(Q_{1}+Q_{2}+2P)$ in (7) we have

$\deg\left(\mathcal{O}_{X}(3)\otimes\mathcal{O}_{X}(Q_{1}+Q_{2}+2P)\right)=3\deg
X+4=3(3e+1)+4=9e+7\,.$

According to the assumptions in the theorem, $g=3e+3\gamma$ and $e\geq
4\gamma+5$, so for the degree of the line bundle
$\mathcal{O}_{X}(3)\otimes\mathcal{O}_{X}(Q_{1}+Q_{2}+2P)$ we obtain
$9e+7>2g-2$. Thus, $\mathcal{O}_{X}(3)\otimes\mathcal{O}_{X}(Q_{1}+Q_{2}+2P)$
is nonspecial, hence

(8)
$h^{0}(X,N_{X/\mathbb{P}^{r}})=h^{0}(X,\mathcal{O}_{X}(3)\otimes\mathcal{O}_{X}(Q_{1}+Q_{2}+2P))+h^{0}(X,\varphi^{\ast}N_{Y/\mathbb{P}^{r-1}}\otimes\mathcal{O}_{X}(P))\,.$

By the Riemann-Roch theorem

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

To compute
$h^{0}(X,\varphi^{\ast}N_{Y/\mathbb{P}^{r-1}}\otimes\mathcal{O}_{X}(P))$ we
use the projection formula, that is,

$\displaystyle
h^{0}(X,\varphi^{\ast}N_{Y/\mathbb{P}^{r-1}}\otimes\mathcal{O}_{X}(P))$
$\displaystyle=h^{0}(Y,{\varphi}_{\ast}(\,{{\varphi}}^{\ast}N_{Y/\mathbb{P}^{r-1}}\otimes\mathcal{O}_{X}(P)\,))$
$\displaystyle=h^{0}(Y,N_{Y/\mathbb{P}^{r-1}}\otimes{\varphi}_{\ast}\mathcal{O}_{X}(P))\,.$

By Proposition 7 we have
$\varphi_{\ast}(\mathcal{O}_{X}(P))\cong\mathcal{O}_{Y}\oplus\mathcal{O}_{Y}(-1)\oplus(\mathcal{O}_{Y}(-2)\otimes\mathcal{O}_{Y}(-Q))$,
so it follows that

$h^{0}(Y,N_{Y/\mathbb{P}^{r-1}}\otimes{\varphi}_{\ast}\mathcal{O}_{X}(P))\\\
=h^{0}(Y,N_{Y/\mathbb{P}^{r-1}})+h^{0}(Y,N_{Y/\mathbb{P}^{r-1}}(-1))+h^{0}(Y,N_{Y/\mathbb{P}^{r-1}}(-2)\otimes\mathcal{O}_{Y}(-Q))\,.$

Since $Y\cong\Gamma$ is general in $\mathcal{M}_{\gamma}$, $\gamma\geq 3$, and
$E$ is a general divisor of degree $e\geq 4\gamma+5$ on $\Gamma$, it follows
by [CLM96, Proposition 2.1 and Proposition 2.12] that
$h^{0}(Y,N_{Y/\mathbb{P}^{r-1}}(-1))=r$ and
$h^{0}(Y,N_{Y/\mathbb{P}^{r-1}}(-2))=0$. The last implies
$h^{0}(Y,N_{Y/\mathbb{P}^{r-1}}(-2)\otimes\mathcal{O}_{Y}(-Q))=0$. Using that
$r-1=e-\gamma$, we find

$\displaystyle
h^{0}(X,{{\varphi}}^{\ast}N_{Y/\mathbb{P}^{r-1}}\otimes\mathcal{O}_{X}(P))$
$\displaystyle=h^{0}(Y,N_{Y/\mathbb{P}^{r-1}})+h^{0}(Y,N_{Y/\mathbb{P}^{r-1}}(-1))$
$\displaystyle=er-(r-4)(\gamma-1)+r$ $\displaystyle=r^{2}+r+4\gamma-4\,.$

The exact sequence (8) then gives
$h^{0}(X,N_{X/\mathbb{P}^{r}})=(6e-3\gamma+8)+(r^{2}+r+4\gamma-4)=r^{2}+r+\gamma+6e+4=r^{2}+(e-\gamma+1)+\gamma+6e+4=r^{2}+7e+5$.
Therefore,

(9)
$\operatorname{dim}T_{[X]}\mathcal{H}=\operatorname{dim}\mathcal{H}+1=r^{2}+7e+5\,.$

This completes Step II.

Step III. Showing that $\mathcal{H}$ forms an irreducible component of
$\mathcal{I}_{d,g,r}$.

By definition, $\mathcal{H}\subset\mathcal{I}_{d,g,r}$ is the closure of the
set parametrizing smooth integral curves of degree $d$ and genus $g$ on cones
in $\mathbb{P}^{r}$ over the curves parametrized by
$\mathcal{I}_{e,\gamma,r-1}$, as a general $[X]\in\mathcal{H}$ is in the
linear equivalence class $3\Gamma_{0}+(3E+q)\mathfrak{f}$ on the
desingularization $S$ of a cone $F\subset\mathbb{P}^{r}$ over
$Y\subset\mathbb{P}^{r-1}$ for a general $[Y]\in\mathcal{I}_{e,\gamma,r-1}$.
The set $\mathcal{H}$ is clearly irreducible. To show that it is a component,
we use that that every flat deformation of a curve from $\mathcal{F}$ is a
again a curve on a cone in $\mathbb{P}^{r}$ over a curve from
$\mathcal{I}_{e,\gamma,r-1}$.

######  Lemma 9.

Let $p_{\mathcal{X}}:\mathcal{X}\to T$ be a flat family of projective curves
in $\mathbb{P}^{r}$ for which there exists a closed point $t_{0}\in T$ such
that:

  1. (i)

$\mathcal{X}_{t_{0}}$ is a smooth integral projectively normal curve of genus
$g=3e+3\gamma$ and degree $3e+1$;

  2. (ii)

$\mathcal{X}_{t_{0}}$ is contained in a cone $F$ over a curve $Y$
corresponding to a general point of $\mathcal{I}_{e,\gamma,r-1}$.

Then there is a neighborhood $U$ of $t_{0}$ in $T$ such that, for all closed
points $t\in U$, $\mathcal{X}_{t}$ is again a curve on a cone over a smooth
integral projectively normal curve of genus $\gamma$ and degree $e$ in
$\mathbb{P}^{r-1}$.

Assuming the validity of Lemma 9, the proof proceeds as follows. Suppose that
$\tilde{X}$ is a flat deformation of $X$. Lemma 9 implies that $\tilde{X}$ is
contained in a cone $\tilde{F}\subset\mathbb{P}^{r}$ over a curve $\tilde{Y}$,
where $[\tilde{Y}]\in\mathcal{I}_{e,\gamma,r-1}$ is general. Let $\tilde{S}$
be the desingularization of $\tilde{F}$ and $\tilde{C}$ be the proper
transform of $\tilde{X}$ on $\tilde{S}$. Proposition 6 implies that
$\tilde{C}\sim 3\tilde{Y}_{0}+(3\tilde{E}+\tilde{q})\tilde{\mathfrak{f}}$,
where $\tilde{\mathfrak{f}}:\tilde{S}\to\tilde{Y}$ is the corresponding
surjective morphism, $\tilde{Y}_{0}$ is the section of minimal self-
intersection, $\tilde{Y}^{2}_{0}=-e$, $\tilde{E}$ is a divisor of $\tilde{Y}$
of degree $e$ such that
$\tilde{S}\cong\mathbb{P}(\mathcal{O}_{\tilde{Y}}\oplus\mathcal{O}_{\tilde{Y}}(-\tilde{E}))$
and $\tilde{q}$ is a point on $\tilde{Y}$. Also, $\tilde{X}$ is the image of a
curve in the linear series
$|3\tilde{Y}_{0}+(3\tilde{E}+\tilde{q})\tilde{\mathfrak{f}}|$ under the
morphism associated to $|\tilde{Y}_{0}+\tilde{E}\tilde{\mathfrak{f}}|$.
Because of the definition of $\mathcal{F}$, the above means that $\tilde{X}$
is a curve from the same family. Therefore $\mathcal{H}$ is a component of
$\mathcal{I}_{3e+1,g,e-\gamma+1}$. The other statements formulated in the
theorem are clear due to the properties of the curves in the family
$\mathcal{F}$ and Proposition 4.

To complete the proof of the theorem it remains to prove the lemma.

###### Proof of Lemma 9.

To a large extent our proof repeats the steps of the proofs of similar
statements given in [Cil87, Proposition 1.6, p.354–356] and [CLM96,
Proposition 4.1, p.176–178]. For this reason, we refer, whenever possible, to
the statements formulated and proved there. The statement is local, so we can
assume that $T=\operatorname{Spec}A$ for a Noetherian ring $A$. Thus, we have
a flat family

$\mathcal{X}\subset\operatorname{Proj}A[x_{0},x_{1},\ldots,x_{r}]=:\mathbb{P}^{r}_{A}\,.$

Since projective normality is an “open property” and $\mathcal{X}_{t_{0}}$ is
supposed to be projectively normal, we can assume further that the family
$\mathcal{X}$ consists of projectively normal curves. By [Har77, Ex. III.9.5,
p.267] the family $\mathcal{X}$ must be _very flat_. In particular, the number
of generators in any degree $n$ of the ideal $I(\mathcal{X}_{t})$ of a curve
$\mathcal{X}_{t}\subset\mathbb{P}^{r}$ from the family is the same for all
$t\in T$. Consider the homogeneous ideal $I(\mathcal{X})$ of $\mathcal{X}$ in
the ring

$R:=A[x_{0},x_{1},\ldots,x_{r}]$

and let $I(\mathcal{X})_{2}$ be the vector space of its elements of degree
two, that is,

$I(\mathcal{X})_{2}:=H^{0}(\mathbb{P}^{r}_{A},\mathcal{I}_{\mathcal{X}}(2))\,,$

where $\mathcal{I}_{\mathcal{X}}$ is the ideal sheaf of $\mathcal{X}$. Take
$J\subset R$ to be the ideal

$J:=\langle I(\mathcal{X})_{2}\rangle\,$

generated by the elements of degree two. Consider the closed subscheme
$\mathcal{W}\subset\mathbb{P}^{r}_{A}$ defined as
$\mathcal{W}:=\operatorname{Proj}(R/J)\subset\mathbb{P}^{r}_{A}\,.$ It is
indeed a family $p_{\mathcal{W}}:\mathcal{W}\to T$ parametrized by
$T=\operatorname{Spec}(A)$ and we have a commutative diagram

${\mathcal{X}\subset\mathcal{W}\subset\mathbb{P}^{r}_{A}}$${\operatorname{Spec}(A)}$$\scriptstyle{p_{\mathcal{X}}}$$\scriptstyle{p_{\mathcal{W}}}$

The goal is to show that $p_{\mathcal{W}}:\mathcal{W}\to
T=\operatorname{Spec}(A)$ is a flat family.

By assumption $\mathcal{X}_{t_{0}}$ is a smooth curve of genus $g=3e+3\gamma$
and degree $3e+1$ contained in a cone $F$ over a smooth integral projectively
normal curve $Y$ of genus $\gamma$ and degree $e$ in $\mathbb{P}^{r-1}$. By
Proposition 6 this means that the proper transform of $\mathcal{X}_{t_{0}}$ on
the desingularization $S$ of $F$ is in the linearly equivalence class of
$3Y_{0}+(3E+q)\mathfrak{f}$, where, just as before, $f:S\to Y$ is the
surjective morphism for the decomposable ruled surface $S$, $Y_{0}$ is the
section of minimal self-intersection, $E$ is a divisor of degree $e$ on $Y$
such that $S\cong\mathbb{P}(\mathcal{O}_{Y}\oplus\mathcal{O}_{Y}(-E))$ and
$q\in Y$ is a point. Since $Y$ is a general curve of genus $\gamma\geq 3$ and
degree $e\geq 4\gamma+5$ in $\mathbb{P}^{r-1}$, it follows by [GL86] that the
first several terms of the minimal free resolution of its ideal sheaf
$\mathcal{I}_{Y}$ appear as

${\cdots}$${\bigoplus\limits^{\beta_{3}}_{j=1}\mathcal{O}_{\mathbb{P}^{r-1}}(-4)}$${\bigoplus\limits^{\beta_{2}}_{j=1}\mathcal{O}_{\mathbb{P}^{r-1}}(-3)}$${\bigoplus\limits^{\beta_{1}}_{j=1}\mathcal{O}_{\mathbb{P}^{r-1}}(-2)}$${\mathcal{I}_{Y}}$${0\,}$

where $\beta_{1},\beta_{2},\ldots$ are the _Betti numbers_. By [CG99,
Proposition 2, p. 232] it follows that the first several terms of the minimal
free resolution of the ideal sheaf $\mathcal{I}_{\mathcal{X}_{t_{0}}}$ of
$\mathcal{X}_{t_{0}}\subset\mathbb{P}^{r}$ are

(10)
${\cdots}$${\mathcal{P}_{3}}$${\mathcal{P}_{2}}$${\mathcal{P}_{1}}$${\mathcal{I}_{\mathcal{X}_{t_{0}}}}$${0\,,}$

where

  * •

$\mathcal{P}_{1}=\bigoplus\limits^{r-1}_{1}\mathcal{O}_{\mathbb{P}^{r}}(-4)\oplus\bigoplus\limits^{\beta_{1}}_{j=1}\mathcal{O}_{\mathbb{P}^{r}}(-2)$

  * •

$\mathcal{P}_{2}=\bigoplus\limits^{\binom{r-1}{2}}_{1}\mathcal{O}_{\mathbb{P}^{r}}(-5)\oplus\bigoplus\limits^{\beta_{2}}_{j=1}\mathcal{O}_{\mathbb{P}^{r}}(-3)\oplus\bigoplus\limits^{\beta_{1}}_{1}\mathcal{O}_{\mathbb{P}^{r}}(-5)$

  * •

$\mathcal{P}_{3}=\bigoplus\limits^{\binom{r-1}{3}}_{1}\mathcal{O}_{\mathbb{P}^{r}}(-6)\oplus\bigoplus\limits^{\beta_{3}}_{j=1}\mathcal{O}_{\mathbb{P}^{r}}(-4)\oplus\bigoplus\limits^{\beta_{2}}_{1}\mathcal{O}_{\mathbb{P}^{r}}(-6)$

To deduce the flatness of the family $p_{\mathcal{W}}:\mathcal{W}\to T$ we
make use of resolutions of the ideal $I(\mathcal{X})\subset R$ of
$\mathcal{X}$, the ideal $I(\mathcal{X}_{t_{0}})$ of $\mathcal{X}_{t_{0}}$ in
the localization $R_{t_{0}}$ of $R$ at $t_{0}$, and also of the ideal $J$ of
$\mathcal{W}\subset\mathbb{P}^{r}_{A}$. Remark that due to (10), the ideal
$I(\mathcal{X}_{t_{0}})$ has a presentation

(11) ${P_{2}}$${P_{1}}$${I(\mathcal{X}_{t_{0}})}$${0\,,}$

where
$P_{1}=\bigoplus\limits^{\beta_{1}}_{j=1}R_{t_{0}}(-2)\oplus\bigoplus\limits^{r-1}_{1}R_{t_{0}}(-4)$
and
$P_{2}=\bigoplus\limits^{\beta_{2}}_{j=1}R_{t_{0}}(-3)\oplus\bigoplus\limits^{\binom{r-1}{2}+\beta_{1}}_{1}R_{t_{0}}(-5)$.
By the result of Catalisano and Gimigliano explained in section 2, the zero
locus of the degree two generators of $I(\mathcal{X}_{t_{0}})$ is precisely
the cone $F$ containing the curve $\mathcal{X}_{t_{0}}$, that is,
$V(J_{t_{0}})\equiv F$, where $J_{t_{0}}$ is the ideal of the fiber
$\mathcal{W}_{t_{0}}$ of $\mathcal{W}$ at the point $t_{0}$. Just like in the
proof of [Cil87, Proposition 1.6] it is obtained that there is a commutative
diagram

(12)
${Q_{2}}$${Q_{1}}$${I(\mathcal{W}_{t_{0}})}$${0}$${P_{2}}$${P_{1}}$${I(\mathcal{X}_{t_{0}})}$${0}$$\scriptstyle{\delta}$$\scriptstyle{\theta}$

with exact rows, where $Q_{1}=\bigoplus\limits^{\beta_{1}}_{j=1}R_{t_{0}}(-2)$
and $Q_{2}=\bigoplus\limits^{\beta_{2}}_{j=1}R_{t_{0}}(-3)$, $\delta$ is
represented by a $\beta_{1}\times\beta_{2}$ matrix of linear forms and
$\theta$ is represented by a matrix $M$ of the form

$M=\begin{pmatrix}M_{1,1}&M_{1,2}\\\ 0&M_{2,2}\end{pmatrix}\,,$

for which $M_{1,1}$ is a $\beta_{1}\times\beta_{2}$ matrix of linear forms,
$M_{1,2}$ is a $\beta_{1}\times\left(\binom{r-1}{2}+\beta_{1}\right)$ matrix
and $M_{2,2}$ is a $(r-1)\times\left(\binom{r-1}{2}+\beta_{1}\right)$ matrix.
As it is explained in [Cil87, Proposition 1.6], because of the very flatness
of the family $p_{\mathcal{X}}:\mathcal{X}\to T$ there is a presentation

(13)
${\mathscr{P}_{2}}$${\mathscr{P}_{1}}$${I(\mathcal{X})}$${0\,,}$$\scriptstyle{\Theta}$

of $I(\mathcal{X})$ by free $R$-modules such that the localization of the
sequence (13) at $k(t_{0}):=A/m_{t_{0}}$ gives (11), where $m_{t_{0}}\subset
A$ is the ideal corresponding to the point $t_{0}\in T$ and $\Theta$ is a
homogeneous map represented by a matrix $\mathscr{M}$ of the form

$\mathscr{M}=\begin{pmatrix}\mathscr{M}_{1,1}&\mathscr{M}_{1,2}\\\
\mathscr{M}_{2,1}&\mathscr{M}_{2,2}\end{pmatrix}\,,$

which modulo the ideal $m_{t_{0}}$ of $t_{0}$ gives the matrix $M$. The same
degree reasoning argument as in [Cil87, Proposition 1.6] gives that
$\mathscr{M}_{2,1}=0$ and one can “chop-off” from (13) an exact sequence

(14) ${\mathscr{Q}_{2}}$${\mathscr{Q}_{1}}$${J}$${0\,,}$$\scriptstyle{\Delta}$

where $\Delta$ is homogeneous map, such that tensoring (14) with $k(t_{0})$ we
get the first row of (12). This means that the corank of the map $\Delta$ at
each localization at $k(t)=A/m_{t}$, where $m_{t}$ is the ideal corresponding
to $t\in T$, is same for all $t$, or equivalently, that
$\operatorname{dim}(J_{t})_{d}$ is same for all $t\in T$. This implies that
the family $p_{\mathcal{W}}:\mathcal{W}\to T$ is (very) flat. In particular,
it is a family of surfaces in $\mathbb{P}^{r}$ one of whose fibers, namely
$\mathcal{W}_{t_{0}}$, is a cone over a smooth integral projectively normal
curve in $\mathbb{P}^{r-1}$ of genus $\gamma\geq 3$ and degree $e\geq
4\gamma+5$.

For the remaining part of the proof of the lemma we refer to [CLM96,
Proposition 4.1, p.176–178]. It is proven there that if
$p_{\mathcal{W}}:\mathcal{W}\to T$ is a flat family of surfaces in
$\mathbb{P}^{r}$, one of whose fibers is a cone like $\mathcal{W}_{t_{0}}$
above, then the remaining fibers of the family are also cones over smooth
curves of the same genus and degree in $\mathbb{P}^{r-1}$. We remark that the
proof uses a result of Pinkham, namely [Pin74, Theorem 7.5, p.45] about cones
over curves of genus $\gamma$ and degree $e$, in which it is required that
$e\geq 4\gamma+5$. Thus the lemma is proved.∎

This completes Step III and thus the proof of the Main Theorem.

######  Remark 10.

The technique used in Step III of the proof can not be applied to prove
[CIK22, Theorem B]. In that paper we considered a family of curves that also
lie on cones such that each curve is a double cover of the base of the cone
that contains it and also passes through its vertex. Proposition 1 can be
applied to obtain a resolution of the ideal of a curve from the family.
However, the ideal is generated by polynomials of degree two and three, which
is insufficient to deduce the existence of a presentation like (14) of the
ideal of a similarly defined variety like $\mathcal{W}$ here. That is, one
couldn’t conclude that $\mathscr{M}_{2,1}=0$ like we were able to do here due
to $I(\mathcal{X}_{t})$ being generated by polynomials of degree two and four.
In a sense, our present work grew-out from our failure to apply some
techniques used in [CLM96] and [Cil87] in the proof of [CIK22, Theorem B, step
III].

######  Remark 11.

For a component $\mathcal{D}$ of the Hilbert scheme $\mathcal{I}_{d,g,r}$ the
difference
$\sigma(\mathcal{D}):=\operatorname{dim}\mathcal{D}-\lambda_{d,g,r}$ is called
_superabundance_. It is not difficult to compute about our
$\mathcal{H}\subset\mathcal{I}_{3e+1,3e+3\gamma,e-\gamma+1}$ that
$\sigma(\mathcal{H})=(r-4)e+2(r-5)(e-r)-3$, and using the numerical
assumptions in our Main Theorem, $\sigma(\mathcal{H})\geq 224$.

## References

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# Investigating Semi-Supervised Learning Algorithms in Text Datasets

1st H. Toprak Kesgin  Department of Computer Engineering
Yildiz Technical University Istanbul, Turkey
<EMAIL_ADDRESS>2nd M. Fatih Amasyali  Department of Computer
Engineering
Yildiz Technical University Istanbul, Turkey
<EMAIL_ADDRESS>
###### Abstract

Using large training datasets enhances the generalization capabilities of
neural networks. Semi-supervised learning (SSL) is useful when there are few
labeled data and a lot of unlabeled data. SSL methods that use data
augmentation are most successful for image datasets. In contrast, texts do not
have consistent augmentation methods as images. Consequently, methods that use
augmentation are not as effective in text data as they are in image data. In
this study, we compared SSL algorithms that do not require augmentation; these
are self-training, co-training, tri-training, and tri-training with
disagreement. In the experiments, we used 4 different text datasets for
different tasks. We examined the algorithms from a variety of perspectives by
asking experiment questions and suggested several improvements. Among the
algorithms, tri-training with disagreement showed the closest performance to
the Oracle; however, performance gap shows that new semi-supervised algorithms
or improvements in existing methods are needed.

###### Index Terms:

semi supervised learning, self-training, co-training, tri-training, tri-
training with disagreement

##  I Introduction

Larger training datasets increase the generalizability of neural networks. If
a large amount of data is used, it often outperforms other traditional methods
of machine learning. However, data collection and labeling are laborious and
costly. It may not always be possible to train neural networks with enough
samples. Sometimes, data is readily available, but labeling can be costly. For
example, labeling text or image data is more expensive than acquiring it.

Semi-supervised learning (SSL) is a machine learning method that uses only a
few labeled data and a relatively large number of labeled data. As its name
suggests, SSL lies between supervised and unsupervised learning. SSL aims to
outperform supervised learning, which uses only labeled samples, and
unsupervised learning, in which no samples are labeled.

SSL is gaining popularity with the aim of reducing the amount of labeled data
required for neural networks. It can be achieved by directly using or adapting
existing methods for neural networks. SSL uses supervised learning techniques
to generate training data.

There are many methods and techniques for SSL. The first of these is the
consistency regularization method [1, 2]. This method assumes that adding
small noise to data points should not have a significant impact on label
estimates. Then it is possible to add unlabeled data points in which the
predictions do not change much to a training dataset.

Some SSL methods use generative models [3, 4, 5]. Using such methods, new data
points are created from the original data points for training purposes.
Methods like these can be applied to datasets where augmentation can be
performed successfully. Although data augmentation techniques provide
significant improvements to image datasets, they do not perform as well with
text datasets.

Proxy-label methods are one of the main SSL techniques that don’t need data
augmentation. [6, 7, 8, 9, 10] The proxy label method relies on predicting the
label of unlabeled data from labeled data. After that, the predicted samples
are combined with the samples with known labels for training. Therefore, extra
information is provided to the model even if the predicted labels are noisy or
do not reflect the ground truth.

Proxy label methods differ according to how the labels are predicted. Self-
training is the training of a model with known-label samples, then using this
model to iteratively incorporate the label predictions of unknown-label
samples into the training. Self training is one of the earliest methods of
SSL. Another proxy label method, co-training, uses two sets of features that
are conditionally independent to predict labels. In other words, two different
models are trained with two different feature sets for the same dataset.
Models provide labels to each other when one’s predictions have a high level
of confidence, while the other’s do not. As a result, training can continue
with the addition of new data points with labels from a different perspective.

Tri-training is another method in proxy label methods. In tri-training, three
different models are trained simultaneously. If the predictions of the two
models are consistent with each other, this label is given to the third model.
Tri-training improves its label prediction accuracy by using multiple models,
but it can be computationally expensive since it trains three different
models.

The purpose of this study was to compare the performance of proxy label
methods that do not require augmentation for text data. We analyze some
situations in the training of algorithms using various questions and answer
them in the experiments section. The performance of the methods was compared
according to their supervised and oracle performances on text datasets for
four different tasks. Supervised learning refers to training using only a
small number of samples with known labels. Oracle, on the other hand, refers
to the state where all instances have labels. Oracle represents the upper
limit of semi-supervised performance. It is the aim of SSL to deliver better
results than supervised learning, as close to the oracle level as possible.

##  II Literature Review

Generative SSL methods usually include generative adversarial networks (GANs)
[4] and variational Auto Encoders (VAEs) [5]. With GANs, real data points can
be used to learn distributions of samples and generate high-quality samples.
Samples produced in this manner can be used directly as labeled for SSL
training or can be used to regularize the SSL training model. As an example,
SGAN modifies the GAN architecture in order to also produce a class label
using the discriminator network [3]. The proposed SGAN architecture improves
classification performance over the non-generative component case.
Additionally, SGAN improves the quality of the generated samples and reduces
the generation time.

$\Pi$ Model, which is one of the consistency regularization methods, creates
two random augmentation methods for labeled and unlabeled data [11]. For
unlabeled data, randomness in augmentation, drop out, and random max pooling
can lead to different predictions. The consistency of these different
estimates establishes the confidence of the $\Pi$ model in unlabeled data.
Other consistency editing methods include Ladder Network[12, 13], Temporal
Ensembling [14], Mean Teacher[15].

One of the oldest SSL methods, self training [6, 7], has shown success in
various text classification tasks like sentiment analysis [16, 17] and part of
speech tagging [18].

Co-training, a method used to train multiple models, has been proven to be
effective in a variety of fields[8, 9]. Several studies have found that the
training algorithm has an impact on co-training performance. There is a
significant performance improvement with co-training when using support vector
machines in the email classification task [19]. Tri-training is similar to co-
training; however, trains three classifiers and provides significant
improvements in the web page classification task and UCI datasets [10]. An
extension of tri-training for neural networks is Tri-net [20]. Authors found
that Tri-net performs significantly on the CIFAR-10 dataset with only a few
labeled samples.

In this paper, we used text datasets and classified them using Bert vectors
and neural networks. SSL methods have been explained and discussed from a
variety of perspectives in the methods section. In the experiments section, we
discussed SSL with asking and answering some unexamined questions.

##  III Methods

We explain in this section the SSL algorithms and their extensions used in our
experiments. Self training is based on combining labeled and unlabeled data.
Unlabeled samples with high confidence scores are included in the training if
the estimation is above a certain threshold. The threshold value is based on
the model prediction’s confidence value, which is a probability value between
0 and 1. When this threshold is high, fewer unlabeled samples are included in
training. Also, due to the fact that the model already predicts with high
confidence, the impact of these samples on the model’s weights is limited.
When this threshold is kept low, incorrectly labeled samples can be included
in the training since the label’s confidence rate is low, and this will
adversely affect the model’s performance.

In most cases, this threshold is a single value and samples higher than a
certain confidence level are selected. However, the threshold value does not
have to be a single value. By including samples that are smaller than a
certain confidence interval, the model’s performance may be improved, since it
will provide differentiation in training. The pseudocode for the threshold-
based self-training algorithm is given in Algorithm-1. In Algorithms 1 and 2,
$D$ represents the labeled dataset, $U$ represents the unlabeled dataset, $m$
represents the model to be trained, and $\tau_{1}$ and $\tau_{2}$ represent
threshold values.

Algorithm 1 Threshold-Based Self-training

1:$m\leftarrow train\\_model(D)$

2:repeat

3: $D_{i}\leftarrow\varnothing$

4: for $x\in U$ do

5: if $\max m(x)>\tau_{1}$ and $\max m(x)<\tau_{2}$ then

6: $D_{i}\leftarrow\\{(x,p(x))\\}$

7: $m\leftarrow train\\_model(D\cup D_{i})$

8:until All samples are included in the training OR max iteration count is
reached

It is also possible to define self-training based on the number of samples
included rather than thresholds. For example, the most reliable x samples can
be included in every iteration. We will call this approach count-based self
training. The pseudocode for count-based self training is found in
Algorithm-2.

Algorithm 2 Count-Based Self-training

1:$m\leftarrow train\\_model(D)$

2:repeat

3: $D_{i}\leftarrow\varnothing$

4: Sort $U$ from greatest to least according to model’s confidence scores

5: $D_{i}$ = $U[th1..th2]$

6: $m\leftarrow train\\_model(D\cup D_{i})$

7:until All samples are included in the training OR max iteration count is
reached

Some algorithms may need to be trained from scratch with new datasets, but
neural networks allows the model to be trained where it left off with new-
different datasets.

Therefore, the train model function in the 7th line of the algorithms allows
you to train a neural network in both directions from the beginning and from
where you left off in the previous model. In the experiments section, we
examined the effects of these two different cases.

Two main differences between co-training and self-training are that co-
training requires two different sets of features for training set and uses two
models for training. These two feature sets must be sufficient to train a
model and must be conditionally independent. Co-training involves two feature
sets and two models, which allows the models to view the dataset from
different perspectives. Co-training includes samples in training that have a
low confidence value for one model and a high confidence value for the other
model. As a result, more information can be added to the model and better
models can be trained. The pseudocode for co-training is found in Algorithm-3.
where $D_{1}$ and $D_{2}$ are two different feature sets for the training set,
$U$ unlabeled dataset with the same features, $\tau$ is the threshold $m_{i}$
is the $i^{th}$ model, $m(x)$ model’s confidence scores for each class, and
$p_{i}(x)$ is the label that was predicted by the $i^{th}$ model for the
sample $x$.

Algorithm 3 Co-training

1:$m_{1}\leftarrow train\\_model(D_{1})$

2:$m_{2}\leftarrow train\\_model(D_{2})$

3:repeat

4: $L_{1}\leftarrow\varnothing$

5: $L_{2}\leftarrow\varnothing$

6: for $x\in U$ do

7: if $\max m_{1}(x)>\tau\>{\bf and}\>\max m_{2}(x)<\tau$ then

8: $L_{1}\leftarrow\\{(x,p_{1}(x))\\}$

9: if $\max m_{2}(x)>\tau\>{\bf and}\>\max m_{1}(x)<\tau$ then

10: $L_{2}\leftarrow\\{(x,p_{2}(x))\\}$

11: $m_{1}\leftarrow train\\_model(D_{1}\cup L_{2})$

12: $m_{2}\leftarrow train\\_model(D_{2}\cup L_{1})$

13:until no more predictions are confident based on _one_ classifier

Three models are simultaneously trained in tri-training. If both models agree
on the label of a data point, then the third model includes that data point in
its training set. Tri-training with disagreement adds the condition that the
third model makes a different prediction than the two other models. Unlike co-
training, tri-training does not require two sets of features. Tri-training
models differentiate models using bootstrap sampling method and randomness in
model training. When two models agree on a sample’s label, it increases
confidence over a single model. After training is complete, the 3 models are
combined to provide a final prediction. The pseudocode for tri-training and
tri-training with disagreement is given in Algorithm 4. Tri learning with
disagreement is indicated by between square brackets ( $[]$ ). where $D$ is
the training set, $U$ is an unlabeled dataset, $m_{i}$ is the $i^{th}$ model,
$m(x)$ model’s confidence scores for each class, and $p_{i}(x)$ is the label
predicted by the $i^{th}$ model for sample $x$.

Algorithm 4 Tri-training [With Disagreement]

1:for $i\in\\{1..3\\}$ do

2: $S_{i}\leftarrow bootstrap\\_sample(D)$

3: $m_{i}\leftarrow train\\_model(S_{i})$

4:repeat

5: for $i\in\\{1..3\\}$ do

6: $D_{i}\leftarrow\emptyset$

7: for $x\in U$ do

8: if $p_{j}(x)=p_{k}(x)(j,k\neq i)$ [ and $p_{i}(x)!=p_{k}(x)$ ] then

9: $D_{i}\leftarrow D_{i}\cup\\{(x,p_{j}(x))\\}$

10: $m_{i}\leftarrow train\\_model(D\cup D_{i})$

11:until none of $m_{i}$ changes

12:apply majority vote over $m_{i}$

##  IV Experiments

###  IV-A Datasets

Neural network’s generalization capacity and performance over a test set
increases as the size of the training set increases. Figure 1 experimentally
illustrates this on the 1150 news[21] dataset.

Figure 1: Effect of dataset size on accuracy

Thus, SSL algorithms have a meaningful effect when there is little training
data. When SSL is applied with enough training data, the increase in accuracy
is very low. This will make it difficult to understand the effects of the
algorithms. For these reasons, we chose datasets with a small number of
training samples or included subsets of large sample datasets. In the
experiments, we used text datasets, and we took into account the variety of
tasks when selecting the datasets. A summary of the datasets can be found in
Table 1.

Turkish movie sentiment dataset (Sent) [22] contains comments and scores for
movies. We reduced this data set by randomly selecting 500 positive 500
negative 1000 comments. Sarcasm Headlines Dataset (Sarcasm) [23, 24] also
reduced into 1000 sample which 500 sarcastic and 500 non-sarcastic samples.
The selected examples were automatically translated into Turkish using Google
Translate.

TABLE I: Datasets summary Dataset | Sample | Class | Task
---|---|---|---
1150 news [21] | 1150 | 5 | News classification
Hate [25] | 1000 | 2 | Hate Speech Classification
Sent [22] | 1000 | 2 | Sentiment Analysis
Sarcasm [23, 24] | 1000 | 2 | Sarcasm detection

In the experiments, each text was represented with 768 length vectors using
the Berturk model [26] and training was performed with these vectors.

###  IV-B Training Settings

In all experiments, the same neural network was used as a classifier. In the
first layer of the neural network, there are 768 neurons, which are the input
dimensions. Two hidden layers, each consist of 16 neurons, are then connected
to this layer. Relu is the activation function of these layers. The last layer
is determined by the number of classes in the dataset. Softmax is the
activation function of the last layer that gives confidence scores for each
class of predictions. Adam [27] algorithm is used while optimizing the models.
Neural networks are stochastic algorithms. Therefore, they may produce
different results on different runs of the same dataset. Different weights and
randomness in the optimizer affect the classifier’s performance. For these
reasons, three-fold cross-validation was used when evaluating the models.
After each fold was created, some part of the unlabeled data set was selected
based on the separation rate. The models for each fold were trained five times
with different weights. The maximum test accuracy achieved by the models
throughout the training was accepted as the success of the relevant training.
For each evaluation, 3-fold times 5 trials 15 trainings were performed for
each model. We used the t-test with $p=0.10$ to test the significance of these
15 results between models. In the tables, Abbreviations as follows; Threshold
Based Self Training (TBST), Count Based Self Training (CBST), Tri-training
(TT), Tri-training with Disagreement (TTWD) and Co-Training (CT). In all
tables, green indicates a statistically significant improvement according to
the supervised algorithm, where only labeled data were used. A red color
indicates a statistically significant decline.

###  IV-C How well semi-supervised algorithms work?

In this section, we compare the models without making any hyperparameter
adjustments. In threshold-based self-training, we included samples with
confidence scores greater than 0.90 from unlabeled samples in the training set
at each iteration. For count-based self-training, we included 100 of the most
reliable unlabeled samples in the training set at each iteration. The same
algorithm 4 is used for Tri-training and Tri-training with disagreement. For
co-training, the feature set was divided into two equal parts. In other words,
the first model used the first 384 features, and the second model used the
last 384 features. Ensemble models were used for the evaluation of tri-
training, tri-training with disagreement, and co-training. In each iteration,
the models were optimized by continuing from where they left off in the
previous iteration. Table 2 shows performance comparisons between algorithms.
Oracle specifies the state where all samples in the training dataset have been
labelled, and the maximum performance limit for algorithms. Supervised, on the
other hand, refers to a situation in which only labeled examples are used, and
unlabeled examples are not included in the training.

TABLE II: Comparison of Algorithms Baselines Unlabeled sample rate $0.95$
---
| News | Sent | Sarcasm | Hate
Oracle | 95.08 | 87.36 | 76.88 | 78.96
Supervised | 86.99 | 81.52 | 65.49 | 72.70
TBST | 83.74 | 81.73 | 64.79 | 71.49
CBST | 85.97 | 80.69 | 65.43 | 71.83
TT | 87.67 | 83.05 | 65.71 | 71.81
TTWD | 90.78 | 82.92 | 66.32 | 73.55
CT | 88.37 | 82.61 | 65.04 | 72.79
Unlabeled sample rate $0.90$
| News | Sent | Sarcasm | Hate
Oracle | 95.08 | 87.36 | 76.88 | 78.96
Supervised | 90.54 | 83.29 | 68.53 | 73.98
TBST | 90.54 | 84.01 | 68.32 | 72.94
CBST | 90.14 | 82.88 | 68.04 | 73.68
TT | 92.99 | 84.72 | 68.13 | 73.09
TTWD | 93.67 | 84.49 | 69.44 | 74.89
CT | 92.33 | 84.39 | 68.11 | 74.53
Unlabeled sample rate $0.80$
| News | Sent | Sarcasm | Hate
Oracle | 95.08 | 87.36 | 76.88 | 78.96
Supervised | 93.64 | 85.19 | 71.12 | 76.21
TBST | 93.77 | 85.51 | 70.83 | 75.98
CBST | 93.11 | 84.63 | 70.56 | 75.60
TT | 94.19 | 85.63 | 70.97 | 76.38
TTWD | 94.54 | 85.47 | 71.12 | 76.36
CT | 93.76 | 85.56 | 71.32 | 76.62

We conducted a t-test between the supervised and semi-supervised algorithms.
Green indicates significant improvements in the supervised algorithm, while
red indicates worsening. Among the algorithms, tri-training with disagreement
is the most promising. When the unlabeled dataset is 90%, significant
improvements are achieved in the two datasets compared to supervised learning.
On the other hand, self training algorithms cannot surpass supervised learning
algorithm.

###  IV-D How do sampling methods affect tri-training?

Tri-training trains 3 models with labeled dataset before iterative learning
begins. It is important for different sampling strategies to differentiate
models in tri-training, which is based on the differentiation of model
predictions. The bootstrap method is used to differentiate the predictions of
these models. In this study, we compared five different sampling strategies
for TT and TTWD. As part of choosing these strategies, we decided how many
samples will be selected based on the number of labeled samples and whether we
will select with or without replacement. We will call $x$ the number of
samples in the labeled dataset.

TABLE III: Comparison of sampling strategies for tri-training TT - Unlabeled
Sample Rate 0.90
---
Size | Replacement | News | Sent | Sarcasm | Hate
x | False | 92.99 | 84.72 | 68.13 | 73.09
2x | True | 93.39 | 84.40 | 68.41 | 74.43
x | True | 92.80 | 84.45 | 68.29 | 73.00
x/2 | True | 88.71 | 84.31 | 66.52 | 70.96
x/3 | No intersection | 83.22 | 84.76 | 67.40 | 70.85
TTWD - Unlabeled Sample Rate 0.90
Size | Replacement | News | Sent | Sarcasm | Hate
x | False | 93.67 | 84.49 | 69.44 | 74.89
2x | True | 93.95 | 84.71 | 69.61 | 74.77
x | True | 93.90 | 84.80 | 69.01 | 74.64
x/2 | True | 93.31 | 84.84 | 68.85 | 74.51
x/3 | No intersection | 93.24 | 85.00 | 68.49 | 74.64

Table 3 presents a comparison of tri-training algorithms for different
sampling strategies when initially training the models. Size is the number of
samples selected; replace is whether replacement is performed with or without
sampling. On the other hand, no intersection means that the data set is
divided into three subsets that do not intersect. The sampling strategies were
used to train the models before the iterative process started. It is a
replacement for the second line of Algorithms 3 and 4. When the results are
examined, it is clear that although the decrease in the number of samples
selected for TT will differentiate the models, it will adversely affect the
initial performance of the models. On the contrary, TTWD does not degrade
performance when the number of selected samples is decreased. The disagreement
condition in this TTWD improves the poor initial performance of the models. It
does not significantly affect the results of either algorithm if the sampling
is done with or without replacement.

###  IV-E Is it better to train a model from scratch every time?

The weights of neural networks are randomly initialized at the beginning of
training. Using SSL algorithms, the trained model is used to predict labels
for unlabeled samples, and some of these samples are added to the training set
based on certain criteria. After unlabeled samples are included in the
training set, the model must be (re)trained. In the previous sections, we
continued training using the same neural network model. However, with the new
samples, it is possible to train a new model with different initial weights.
The purpose of this section is to examine the effect of randomly initializing
the weights of neural networks during SSL training. In other words, it is a
replacement for the _train_model_ function in the algorithms in the methods
section.

TABLE IV: Effect of creating a new model at each iteration Unlabeled Sample
Rate 0.90
---
Alg | New Model | News | Sent | Sarcasm | Hate
TBST | True | 93.98 | 85.77 | 69.84 | 74.49
TBST | False | 90.54 | 84.01 | 68.32 | 72.94
CBST | True | 92.80 | 85.19 | 70.24 | 75.19
CBST | False | 90.14 | 82.88 | 68.04 | 73.68
TT | True | 93.67 | 85.31 | 68.97 | 74.38
TT | False | 92.99 | 84.72 | 68.13 | 73.09
TTWD | True | 92.40 | 84.60 | 68.77 | 74.70
TTWD | False | 93.67 | 84.49 | 69.44 | 74.89
CT | True | 93.39 | 85.37 | 69.19 | 74.85
CT | False | 92.33 | 84.39 | 68.11 | 74.53

The performance of self-training improved significantly by training a new
model in each iteration. Two datasets showed statistically significant
improvements, while there were no significant improvements before. In other
algorithms, significant improvements in performance were not seen when
compared to the new model was not trained. Even though training models from
scratch improves performance in some algorithms, it is very costly in terms of
training time.

###  IV-F Is it better to use an ensemble model or the best single model?

The performance of the tri-training algorithms and co-training are evaluated
by combining the predictions of the models. For these multi-model algorithms,
we investigated whether the most successful single model or the ensemble of 3
models was more successful.

TABLE V: Comparison of ensemble and single most successful model Unlabeled
Sample Rate 0.90
---
Alg | Eval | News | Sent | Sarcasm | Hate
TT | Ensemble | 92.99 | 84.72 | 68.13 | 73.09
TT | Single | 93.34 | 85.07 | 68.72 | 73.90
TTWD | Ensemble | 93.67 | 84.49 | 69.44 | 74.89
TTWD | Single | 93.41 | 84.60 | 69.65 | 75.02
CT | Ensemble | 92.33 | 84.39 | 68.11 | 74.53
CT | Single | 90.75 | 83.68 | 67.41 | 73.87

The single most successful model in tri-training showed slight improvements
over the ensemble model. In tri-training with disagreement, the performance
changes based on the dataset. In contrast, the ensemble model clearly
performed better in co-training.

###  IV-G How different threshold values affect self training?

In self training the model is trained using labeled data first. This model
makes predictions on the unlabeled dataset, and those whose confidence scores
exceed a certain threshold value are trained together with the labeled ones.
The threshold value affects the performance of self-training. Although a high
threshold increases the reliability of the labels, it will not change the
current weights of the models much. By contrast, choosing a low threshold
reduces label reliability. By limiting the threshold from the top, the model
will be able to change its current state with greater margins.

TABLE VI: Comparison of sampling strategies for tri-training Threshold Based
Self Training - Unlabeled Sample Rate 0.90
---
Th1 | Th2 | News | Sent | Sarcasm | Hate
0.7 | 1.0 | 90.45 | 84.08 | 68.08 | 72.60
0.8 | 1.0 | 90.57 | 83.95 | 68.27 | 72.15
0.9 | 1.0 | 90.54 | 84.01 | 68.32 | 72.94
0.7 | 0.9 | 91.86 | 83.64 | 68.65 | 73.41
0.7 | 0.8 | 92.23 | 83.53 | 68.69 | 73.70
0.8 | 0.9 | 91.84 | 83.69 | 68.67 | 73.45
Count Based Self Training - Unlabeled Sample Rate 0.90
Th1 | Th2 | News | Sent | Sarcasm | Hate
0 | 300 | 90.64 | 82.84 | 68.03 | 73.94
0 | 200 | 90.68 | 82.92 | 68.12 | 73.70
0 | 100 | 90.14 | 82.88 | 68.04 | 73.68
100 | 200 | 90.73 | 82.99 | 68.16 | 73.77
100 | 300 | 90.66 | 83.12 | 68.17 | 73.87
200 | 300 | 90.59 | 83.16 | 68.17 | 73.74

In Table 6, we compare two different self-training algorithms and different
threshold values for self-training. Generally, threshold-based self training
outperforms count-based self training. Additionally, instead of only including
samples below a certain confidence level in the table, not including samples
below a certain confidence level provides various improvements, as it creates
differences in model training.

## Conclusions

We compared SSL algorithms that do not require data augmentation across a
variety of text datasets. We discuss the algorithms from various perspectives.
In the experiments section, we examined the effects of trade-offs of
algorithms with different questions. We have proposed several improvements to
some algorithms. Our findings are summarized as follows.

  * •

Although SSL methods have been able to increase success in some datasets, they
have not been effective in all cases.

  * •

Tri-training with disagreement is the most promising among SSL algorithms.

  * •

A small sample size reduces the success rate compared with a large sample size
in initial training of tri-training.

  * •

Despite its higher cost, the success rate of self-training increases
significantly when a new model is trained from scratch for each iteration.

  * •

Instead of one, the use of two thresholds that limit the confidence value at
the top and bottom increases the effectiveness of self-training.

  * •

In tri-training, the best single model and the best ensemble of three models
are no difference in success.

Tri-training with disagreement is the most promising, and algorithms that use
more than one model are generally more effective. However, it seems that due
to a difference between existing methods and Oracle’s performance, in which
the label of all unlabeled samples is correctly known, new SSL algorithms or
improvements in existing methods are waiting to be found.

## Acknowledgment

This study was supported by the Scientific and Technological Research Council
of Turkey (TUBITAK) Grant No: 120E100.

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