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

By Riemann-Roch,

$h^{0}({C_{\mathcal{E}}},\mathcal{O}(2))=28-18+1+h^{0}(K_{C_{\mathcal{E}}}\otimes\mathcal{O}_{C_{\mathcal{E}}}(-2)).$

Note that

$K_{S}\otimes\mathcal{O}_{S}({C_{\mathcal{E}}})\otimes\mathcal{O}_{S}(-2)=|(10;4,3^{4})-3(3;1^{5})|=|(1;1,0^{4})|=|l-e_{1}|$
(23)

and by adjunction we have

$(K_{S}\otimes\mathcal{O}_{S}({C_{\mathcal{E}}})\otimes\mathcal{O}_{S}(-2))\otimes\mathcal{O}_{C_{\mathcal{E}}}=K_{C_{\mathcal{E}}}\otimes\mathcal{O}_{C_{\mathcal{E}}}(-2).$

The restriction map

$\rho:H^{0}(S,K_{S}\otimes\mathcal{O}_{S}({C_{\mathcal{E}}})\otimes\mathcal{O}_{S}(-2))\rightarrow
H^{0}({C_{\mathcal{E}}},K_{C_{\mathcal{E}}}\otimes\mathcal{O}_{C_{\mathcal{E}}}(-2))$

is surjective since
$h^{1}(S,K_{S}\otimes\cal{O}_{S}(-2))=h^{1}(S,\cal{O}(-3(3;1^{5})))=0$. By
(23), we see that
$|K_{C_{\mathcal{E}}}\otimes\mathcal{O}_{C_{\mathcal{E}}}(-2)|$ is cut out on
${C_{\mathcal{E}}}$ by lines through the ordinary four-fold point on the plane
model of ${C_{\mathcal{E}}}$, thus we have

$h^{0}(K_{C_{\mathcal{E}}}\otimes\mathcal{O}_{C_{\mathcal{E}}}(-2))=2.$

Hence $h^{0}({C_{\mathcal{E}}},\mathcal{O}_{C_{\mathcal{E}}}(2))=13$ and the
claim follows.

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

$h^{0}({C_{\mathcal{E}}},N_{{C_{\mathcal{E}}}|\mathbb{P}^{4}})=h^{0}({C_{\mathcal{E}}},N_{S|\mathbb{P}^{4}}\otimes\mathcal{O}_{C_{\mathcal{E}}})+h^{0}({C_{\mathcal{E}}},N_{{C_{\mathcal{E}}}|S})=57,$

and hence

$\dim\mathcal{H}\leq
h^{0}({C_{\mathcal{E}}},N_{{C_{\mathcal{E}}}|\mathbb{P}^{4}})=57.$

By duality it follows that

$\displaystyle\dim{\widetilde{\mathcal{F}}_{2}}$
$\displaystyle=\dim{\widetilde{\mathcal{F}}_{2}}^{\vee}=\dim\cal{H}-\dim\operatorname{Aut}(\mathbb{P}^{4})$
$\displaystyle\leq
h^{0}({C_{\mathcal{E}}},N_{{C_{\mathcal{E}}}|\mathbb{P}^{4}})-\dim\operatorname{Aut}(\mathbb{P}^{4})=33=\dim\mathcal{F}_{2},$

where
$\widetilde{\mathcal{F}}_{2}\subset\mathcal{G}_{\mathcal{L}}\subset\mathcal{G}^{7}_{20}$
is the component containing the irreducible family $\mathcal{F}_{2}$.

Conclusion: ${\widetilde{\cal{F}}_{2}}$ is an irreducible component of
$\widetilde{\cal{G}}{L}\subset\cal{G}^{7}_{20}$ different from the component
containing the irreducible family $\cal{F}_{1}$, from which the reducibility
of $\mathcal{H}^{\mathcal{L}}_{2r+6,r+11,r}$ for $r=7$ follows. ∎

Leaving the unsettled cases $3\leq r\leq 6$ behind for the irreducibility of
$\cal{H}_{2r+6,r+11,r}$, we close this section with a further reducibility
result for the case $g=r+12$. Note that for $g=r+12$,
$\mathcal{H}^{\mathcal{L}}_{g+r-5,g,r}\neq\emptyset$ for every $r\geq 3$.
Compared with our previous irreducibility results, the following result is
more descriptive and extensive in a certain sense, if not better.

######  Theorem 6.3.

$\mathcal{H}^{\mathcal{L}}_{g+r-5,g,r}$, $g=r+12$ is reducible for $9\leq
r\leq 14$. For $r\geq 15$, every component of
$\mathcal{H}^{\mathcal{L}}_{2r+7,r+12,r}$ sits over a component of
$\mathcal{X}_{3,1}$, the Hurwitz space of triple coverings of elliptic curves.

###### Proof.

We present only essential ingredients of proofs, avoiding repetitions of
arguments similar to those we used so far. For a general
$\mathcal{D}\in\mathcal{G}\subset\widetilde{\mathcal{G}}_{\mathcal{L}}\subset\mathcal{G}^{r}_{2r+7}$,
we assume $\mathcal{E}=\mathcal{D}^{\vee}=g^{4}_{15}=g^{4}_{e}$ is base-point-
free.999Note that determining the irreducibility is our main concern, even
though we do not exclude the possibility for $\cal{E}$ having non-empty base
locus. We distinguish two cases.

  1. (i)

$\mathcal{E}$ is birationally very ample. Note that if

$\pi_{1}(e,4)=21\lneq g=r+12\leq\pi(e,4)=26\Longleftrightarrow 10\leq r\leq
14$

the image curve $C_{\mathcal{E}}\subset\mathbb{P}^{4}$ induced by
$\mathcal{E}$ lies on a cubic surface $S\subset\mathbb{P}^{4}$. Therefore, as
we did in the case $r+10\leq g\leq r+11$, we expect that the the
irreducibility (or reducibility in some cases) would follow by focusing on
(possibly singular) curves lying on a scroll and studying the Severi variety
of nodal curves on $S$.

  2. (ii)

$\mathcal{E}$ is compounded. Since $\mathcal{E}$ have empty base locus,
$\cal{E}$ induces a triple covering
$C\stackrel{{\scriptstyle\varphi}}{{\rightarrow}}E\subset\mathbb{P}^{4}$ onto
a smooth elliptic curve $E$, i,e, $\mathcal{E}=\varphi^{*}(g^{4}_{5})$. Recall
that $\mathcal{E}^{\vee}$ is very ample if $r\geq 10$ by Lemma 2.11.

In the range $10\leq r\leq 14$, we will show that there exists at least one
component for which $\mathcal{E}$ birationally very ample. On the other hand,
on a triple covering $C\stackrel{{\scriptstyle\varphi}}{{\rightarrow}}E$ of an
elliptic curve $E$, the residual series of $\varphi^{*}(g^{4}_{5})$ is very
ample. Hence there are at least two candidates of irreducible families which
may form components in the range $10\leq r\leq 14$.

For $r\geq 15$, $\cal{E}$ is compounded by Castelnuovo genus bound and hence
$\cal{E}$ only induces a triple covering of an elliptic curve. Consequently,
there corresponds an irreducible family of smooth curves with
$(d,g,r)=(2r+7,r+12,r)$ over each irreducible component of the Hurwitz space
$\mathcal{X}_{3,1}\subset\mathcal{M}_{g}$, whose irreducibility is not known.
Therefore if $r\geq 15$, the Hilbert scheme
$\mathcal{H}^{\mathcal{L}}_{2r+7,r+12,,r}$ has the same number of components
as $\mathcal{X}_{3,1}$.

Suppose $\mathcal{E}$ is birationally very ample. Assume that
$p_{a}(C_{\mathcal{E}})=\pi(15,4)=26$. Solving (2), (3) we have

$C_{\mathcal{E}}\in\cal{N}_{1}:=|5H|=|5(e+2f)|=|10f+5e|=|10l-5e|=(10;5).$

Let ${\widetilde{C}_{\cal{E}}}$ be the proper transformation of
$C_{\mathcal{E}}\subset S$ under the blowing up $S\cong\mathbb{P}^{2}_{1}$ at

$\delta(p_{a}(C_{\mathcal{E}})):=p_{a}(C_{\mathcal{E}})-g=p_{a}(C_{\mathcal{E}})-(r+12)=14-r$

nodal points. On $\mathbb{P}^{2}_{15-r}$ we have

${\widetilde{C}_{\cal{E}}}\in(10,5,2^{14-r})$

and

$\displaystyle\cal{E}^{\vee}$
$\displaystyle=|K_{\mathbb{P}^{2}_{15-r}}+{\widetilde{C}_{\cal{E}}}-\widetilde{H}|_{|{\widetilde{C}_{\cal{E}}}}$
(24)
$\displaystyle=|(-3;-1^{15-r})+(10;5,2^{14-r})-(2,1,0^{14-r})|_{|{\widetilde{C}_{\cal{E}}}}=(5;3,1^{14-r})_{|{\widetilde{C}_{\cal{E}}}}$

which is very ample by [13]. We also have

${\widetilde{C}_{\cal{E}}}\cdot\cal{E}^{\vee}=(5;3,1^{14-r})\cdot(10,5,2^{14-r})=2r+7.$

Therefore we have an irreducible family of nodal curves on the Hirzebruch
surface $\mathbb{F}_{1}$ – parametrized by the irreducible Severi variety
$\Sigma_{\mathcal{N}_{1},14-r}$ – whose residual curve (i.e. the curve induced
by $\mathcal{E}^{\vee}$) is a smooth, linearly normal curve with
$(d,g)=(2r+7,r+12)$ in $\mathbb{P}^{r}$. Set

$\displaystyle\mathcal{G}_{\cal{N}_{1}}:=$
$\displaystyle\\{~{}\mathcal{E}^{\vee}~{}|~{}\mathcal{E}=g^{4}_{15}~{}\textrm{is
birationally very
ample},C_{\mathcal{E}}\in\Sigma_{\mathcal{N}_{1},14-r},\,p_{a}(C_{\mathcal{E}})=\pi(15,4)\\}$
$\displaystyle\hskip
20.0pt\subset\mathcal{G}\subset\widetilde{\mathcal{G}}_{\mathcal{L}}\subset\mathcal{G}^{r}_{2r+7}.$

By our preceding discussion together with a usual dimension count,

$\displaystyle\dim\mathcal{G}_{\cal{N}_{1}}$
$\displaystyle=\dim\Sigma_{\cal{N}_{1},14-r}-\dim\operatorname{Aut}(S)=30+r$
$\displaystyle>\lambda(2r+7,r+12,r)=40-r.$ (25)

Set

$\mathcal{G}_{3,1}:=\\{~{}\cal{E}^{\vee}~{}|~{}\cal{E}=\varphi^{*}(g^{4}_{5}),~{}C\stackrel{{\scriptstyle\varphi}}{{\rightarrow}}E,~{}\deg\varphi=3,E\mathrm{~{}is~{}elliptic}\\}\subset\widetilde{\mathcal{G}}_{\mathcal{L}}\subset\mathcal{G}^{r}_{2r+7}.$

We have101010 The Hurwitz space $\cal{X}_{n,\gamma}\subset\cal{M}_{g}$ which
is the locus of smooth curves of genus $g$ which are degree $n$ ramified
coverings of smooth genus $\gamma$ curves is of pure dimension
$2g+(2n-3)(1-\gamma)-2$; cf. [4, Theorem 8.23, p. 828].

$\dim\mathcal{G}_{3,1}=\dim\mathcal{X}_{3,1}+\dim
W^{4}_{5}(E)=\dim\mathcal{X}_{3,1}+\dim J(E)=2g-1=2r+23.$

Recall that if $10\leq r\leq 14$, we have either

$\cal{D}=\cal{E}^{\vee}\in\mathcal{G}_{\cal{N}_{1}}\textrm{ or
}\cal{D}=\cal{E}^{\vee}\in\cal{G}_{3,1}.$

We also have $\dim\mathcal{G}_{\cal{N}_{1}}<\dim\cal{G}_{3,1}$. By recalling
that a birationally very ample linear series is not a specialization of
compounded linear series’, $\mathcal{G}_{\cal{N}_{1}}$ and each components of
$\cal{G}_{3,1}$ are distinct components of
$\widetilde{\mathcal{G}}_{\mathcal{L}}\subset\mathcal{G}^{r}_{2r+7}$. If
$r=9$, there is no guarantee that there may exist irreducible families of
smooth curves with $(d,g)=(2r+7,r+12)$ sitting over
$\cal{X}_{3,1}\subset\cal{M}_{g}$. Furthermore,
$C_{\mathcal{E}}\subset\mathbb{P}^{4}$ may not be lie only in a surface of
minimal degree; $\pi_{1}(e,4)=21=g=r+12$ if $r=9$. Instead $C_{\mathcal{E}}$
may lie on a quartic surface unlike the cases $10\leq r\leq 14$. However we
may overcome this subtle situation as follows, which we only give outlines.

  * •

We assume $\cal{E}=\cal{D}^{\vee}$ is base point free and birationally very
ample.

  * •

If $C_{\mathcal{E}}\subset\mathbb{P}^{4}$ lies on a smooth cubic scroll $S$,
we retain the notation $\cal{N}_{1}:=|5H|$.

  * •

In this case we may assume $C_{\mathcal{E}}$ is nodal and is general in
$\Sigma_{\mathcal{N}_{1,14-r}}$. One easily checks $\cal{E}^{\vee}$ is very
ample as we did in (24). Set

$\cal{F}:=\Sigma_{\mathcal{N}_{1,14-r}}/\operatorname{Aut}(S)\subset\cal{G}^{4}_{15}.$

  * •

By (6), $\lambda(d,g,r)=31<30+r=39=\dim\cal{F}$, hence the residual series’ of
the corresponding family $\cal{F}$ may well contribute to a component of
$\mathcal{H}^{\mathcal{L}}_{2r+7,r+12,r}$.

  * •

To show that there exists another component different from $\cal{F}^{\vee}$,
we seek for a suitable irreducible family of curves on smooth del Pezzo in
$\mathbb{P}^{4}$.

  * •

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

$\deg
C_{\mathcal{E}}=3a-\sum^{5}_{i=1}b_{i}=15,~{}C_{\mathcal{E}}^{2}=a^{2}-\sum^{5}_{i=1}b_{i}^{2}=2g-2-K_{S}\cdot
C_{\mathcal{E}}=55$

we get

$(a;b_{1},\cdots,b_{5})=(10;3^{5})\textrm{ or }(11,4^{3},3^{2}),(12;5,4^{4}).$

Note that all the above three linear systems are very ample. Furthermore, a
general member in one of three is isomorphic to a general member of others via
obvious quadratic transformations.

  * •

Let $C_{\mathcal{E}}\in(10;3^{5})$. There is an irreducible family
$\mathcal{F}_{1}\subset\mathcal{G}^{4}_{15}$ consisting of hyperplane series
$\cal{E}$ of $C_{\mathcal{E}}\subset\mathbb{P}^{2}_{5}\cong
S\subset\mathbb{P}^{4}$. Since

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

is very ample, $\mathcal{E}^{\vee}$ induces an embedding into $\mathbb{P}^{r}$
where $r=\dim(4;1^{5})=9$.

  * •

A naive dimension count yields

$\dim\mathcal{F}_{1}=\dim(10;3^{5})+\dim\mathbb{G}(1,14)-\dim\operatorname{Aut}(\mathbb{P}^{4})=37.$

  * •

So far we have the following two irreducible families inside
$\mathcal{G}^{4}_{15}$ which have dimension greater than
$\lambda(15,21,4)=\lambda(25,21,9)=31$.

    1. (1)

$\mathcal{F}$: $\dim\mathcal{F}=39$, $C_{\mathcal{E}}\subset\mathbb{P}^{4}$
induced by $\mathcal{E}\in\mathcal{F}$ is singular lying on a rational normal
scroll $S$ and $C_{\mathcal{E}}$ is $5$-gonal. $\mathcal{E}^{\vee}$ is very
ample for a general $\mathcal{E}\in\mathcal{F}$.

    2. (2)

$\mathcal{F}_{1}$: $\dim\mathcal{F}_{2}=37$,
$C_{\mathcal{E}}\subset\mathbb{P}^{4}$ induced by
$\mathcal{E}\in\mathcal{F}_{1}$ is smooth lying on a smooth del Pezzo surface
and $C_{\mathcal{E}}$ is $7$-gonal.111111 $C_{\mathcal{E}}\in(10;3^{5})$ being
$7$-gonal is not totally clear. One may consult [31, Theorem 2, 3].
$\mathcal{E}^{\vee}$ is very ample for a general
$\mathcal{E}\in\mathcal{F}_{1}$.

  * •

If the Hilbert scheme
$\mathcal{H}^{\mathcal{L}}_{2r+7,r+12,r}=\mathcal{H}^{\mathcal{L}}_{25,21,9}$
is irreducible, then both $\mathcal{F},\mathcal{F}_{1}$ would be contained in
another irreducible family of dimension strictly bigger than
$\dim\mathcal{F}=39$.

  * •

We may argue that this is not the case by showing that $\mathcal{F}_{1}$ is
dense in a component $\tilde{\mathcal{F}_{1}}\subset\mathcal{G}^{4}_{15}$ and
hence $\dim\tilde{\mathcal{F}_{1}}=37$.

  * •

Let $C:=C_{\mathcal{E}}\subset S\subset\mathbb{P}^{4}$, where $C\in(10;3^{5})$
and $S$ is a smooth del Pezzo surface in $\mathbb{P}^{4}$. Let
$c\in\mathcal{H}^{\mathcal{L}}_{15,21,4}$ be a point corresponding to $C$ and
let $\mathcal{H}$ be a component containing $c$. We compute

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

  * •

From the the standard exact sequence

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

it is easy to check

  * •

(i) Claim: $h^{0}(C,N_{C|S})=35$ and $h^{1}(C,N_{C|S})=0.$ (ii) Claim:

$\displaystyle h^{0}(C,N_{{S|\mathbb{P}^{4}}_{|C}})$
$\displaystyle=h^{0}(C,N_{S|\mathbb{P}^{4}}\otimes\mathcal{O}_{C})$
$\displaystyle=h^{0}(C,(\mathcal{O}_{S}(2)\oplus\mathcal{O}_{S}(2))\otimes\mathcal{O}_{C})$
$\displaystyle=2h^{0}(C,\mathcal{O}_{C}(2))=26;$

By Riemann-Roch, we have

$h^{0}(C,\mathcal{O}(2))=30-21+1+h^{0}(K_{C}\otimes\mathcal{O}_{C}(-2)).$

Note that

$K_{S}\otimes\mathcal{O}_{S}(C)\otimes\mathcal{O}_{S}(-2)=|(10;3^{5})-3(3;1^{5})|=|(1;0^{5})|=|l|$

and by adjunction

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

we see that $|K_{C}\otimes\mathcal{O}_{C}(-2)|$ is cut out on $C$ by lines in
the plane model of $C$ and therefore

$h^{0}(K_{C}\otimes\mathcal{O}_{C}(-2))=3.$

Hence $h^{0}(C,\mathcal{O}_{C}(2))=13$ and Claim (ii) follows.

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

$h^{0}(C,N_{C|\mathbb{P}^{4}})=h^{0}(C,N_{S|\mathbb{P}^{4}}\otimes\mathcal{O}_{C})+h^{0}(C,N_{C|S})=61,$

and therefore

$\dim\mathcal{H}\leq h^{0}(C,N_{C|\mathbb{P}^{4}})=61.$

It finally follows that

$\dim\tilde{\mathcal{F}}_{1}\leq
h^{0}(C,N_{C|\mathbb{P}^{4}})-\dim\operatorname{Aut}(\mathbb{P}^{4})=37,$

thus $\mathcal{F}_{1}$ is dense in $\tilde{\mathcal{F}}_{1}$. Passing to the
residual family $\cal{F}_{1}^{\vee}\subset\cal{G}^{9}_{25}$ we see that
$\mathcal{H}^{\mathcal{L}}_{25,21,9}$ has a component of the expected
dimension other than the one arising from the family $\cal{F}^{\vee}$.

∎

######  Remark 6.4.

The family of simple triple coverings of a fixed elliptic curve is irreducible
by a work of Kanev; cf. [20]. Therefore it is likely that if $r\geq 15$,
$\mathcal{H}^{\mathcal{L}}_{2r+7,r+12,r}$ is irreducible, for which the author
does not have any proper justification at this moment.

##  7\. An epilogue, beyond index of speciality five

In this final section, we would like to discuss and make a couple of
statements on families of projective curves with arbitrarily given index of
speciality $\alpha$, as an attempt to seek for a reasonable generalization of
the results we obtained for $\alpha\leq 5$. Even though we need to impose
certain restrictions on the range of $(d,g,r)$ in which the statements work,
it may be worthy of coming up with some generalizations for further study
along the line we followed in this paper. For almost all the statements which
will be made in this section, the author is reluctant to provide proofs,
mainly because this article would get too lengthy. Verifications for those
statements and more extensive relevant results can be found in our forthcoming
paper [24].

We are interested in studying Hilbert schemes of smooth curves of degree $d$
and genus $g$ in $\mathbb{P}^{r}$ with a given fixed index of speciality.
There is an upper bound of the index of speciality of the hyperplane series of
a linearly normal projective curve in terms of $(d,g)$. To be precise, we
recall the following.

######  Remark 7.1.

  * (i)

By Castelnuovo theory, it is known that the dimension $r$ of a birationally
very ample linear series $g^{r}_{d}$ on a curve of genus $g$ is bounded by
(cf. [26, Lemma1.5])

$r\leq\begin{cases}\frac{d+1}{3}&~{}\mathrm{~{}if~{}}d\leq g\\\
\frac{1}{3}(2d-g+1)&~{}\mathrm{~{}if~{}}d\geq g.\end{cases}$ (26)

  * (ii)

Setting $g-d+r=\alpha$, the bound (26) can be read as

$\alpha\leq\begin{cases}\frac{3g-2d+1}{3}&~{}\mathrm{~{}if~{}}d\leq g\\\
\frac{1}{3}(2g-d+1)&~{}\mathrm{~{}if~{}}d\geq g.\end{cases}$ (27)

We recall Claim (3.2.A) in the proof of Theorem 3.2, which we quote as the
following proposition.

######  Proposition 7.2.

Given $\alpha\geq 3$, we have

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

if

$r\geq\alpha+1\mathrm{~{}~{}and~{}~{}}g\geq r+3\alpha-2.$

######  Remark 7.3.

  * (i)

One of numerical condition $r\geq\alpha+1$ in Proposition 7.2 is equivalent to
$d\geq g+1$. One may think that imposing the restriction $r\geq\alpha+1$ is a
too much sacrifice in view of the fact that many hard but interesting cases
occur when $d\leq g$. On the contrary, our knowledge on the Hilbert scheme
$\mathcal{H}^{\mathcal{L}}_{d,g,r}$ is still on the poor side even when
$r\geq\alpha+1$ as far as the author perceives. Therefore it is worthwhile to
settle down or at least make a comprehensive treatment under the assumption
$r\geq\alpha+1$. Accordingly, for the rest of this section, we almost always
assume $r\geq\alpha+1$ unless otherwise specified. We also assume $\alpha\geq
6$ simply because we treated $\alpha=5$ thoroughly in preceding sections.

  * (ii)

We note that the bound (27) is sharp. Indeed, when the index of speciality
attains its maximal possible value $\alpha=\frac{1}{3}(2g-d+1)$ – the
condition equivalent to the numerical condition $r=\frac{1}{3}(2d-g+1)$ in
(26) – a curve $C\subset\mathbb{P}^{r}$ with maximal index of speciality
$\alpha$ is an extremal curve; by direct computation, we have in this case

$g=r+2\alpha-1=\pi(d,r).$

  * (iii)

If $g\leq r+2\alpha-2$ – which is equivalent to the condition
$r\gneq\frac{1}{3}(2d-g+1)$ – we get a contradiction by (26). Therefore we
arrive at the following statement which can be viewed as a generalization of
Theorem 3.2 (a), (b) and Propostion 4.1 (i), (ii).

######  Proposition 7.4.

Suppose $r\geq\alpha+1$ and $\alpha\geq 6$. We have

  * (i)

$\cal{H}_{d,g,r}=\mathcal{H}^{\mathcal{L}}_{d,g,r}=\emptyset\mathrm{~{}for~{}}g\leq
r+2\alpha-2$

  * (ii)

$\cal{H}_{d,g,r}=\mathcal{H}^{\mathcal{L}}_{d,g,r}\neq\emptyset\mathrm{~{}for~{}}g=r+2\alpha-1$
and is reducible only if $r=\alpha+1$.

From Proposition 7.4 and Proposition 7.2, it is natural to ask whether

$\mathcal{H}^{\mathcal{L}}_{d,g,r}\neq\emptyset\mathrm{~{}~{}\hskip
12.0ptfor~{}~{}\hskip 12.0pt}r+2\alpha\leq g\leq r+3\alpha-3.$

When $g=r+2\alpha$ or $g=r+2\alpha+1$, one can prove the following which is a
partial generalization of Theorem 3.2 (c), (d).

######  Proposition 7.5.

Fix $\alpha\geq 5$:

  * (a)

$\mathcal{H}^{\mathcal{L}}_{d,g,r}=\emptyset$ for $g=r+2\alpha$ if
$r\geq\alpha+4$.

  * (b)

$\mathcal{H}^{\mathcal{L}}_{d,g,r}=\emptyset$ for $g=r+2\alpha+1$ if
$r\geq\max\\{\alpha+6,12\\}$.

The converse of Proposition 7.5, namely

(a$\dagger$) $\mathcal{H}^{\mathcal{L}}_{d,g,r}\neq\emptyset$ for
$g=r+2\alpha$ and for every $3\leq r\leq\alpha+3$

(b$\dagger$) $\mathcal{H}^{\mathcal{L}}_{d,g,r}\neq\emptyset$ for
$g=r+2\alpha+1$ and for every $3\leq r\leq\alpha+5$

has not been completely settled yet if $\alpha\geq 6$. Some partial results
have been obtained which does not seem to be adequate to be stated precisely
here.

When the genus $g$ is still small but not too small with respect to $\alpha$
and $r$, say when $g=r+2\alpha+2$, one has the following result which
partially recovers Theorem 3.2(e) as well as Theorem 6.3.

######  Proposition 7.6.

Fix $\alpha\geq 6$. We have

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

for $g=r+2\alpha+2$ if $r\geq\alpha+5$. Furthermore, in case
$g=r+2\alpha+2\geq\max\\{3\alpha+10,2\alpha+17\\}$, a general element of any
component of

$\mathcal{H}^{\mathcal{L}}_{d,g,r}=\mathcal{H}^{\mathcal{L}}_{g+r-\alpha,g,r}=\mathcal{H}^{\mathcal{L}}_{2r+\alpha+2,r+2\alpha+2,r}$

are triple covers of elliptic curves.

######  Remark 7.7.

  * (i)

If $g\geq r+2\alpha+3$, the existence of linearly normal smooth curves in
$\mathbb{P}^{r}$ with index of speciality $\alpha$ is only assured for
$\alpha=5$ as we have seen in Theorem 3.2 (e). Note the condition $g\geq
r+3\alpha-2$ we had in Proposition 7.2 is same as $g\geq r+2\alpha+3$ if
$\alpha=5$. However, for $\alpha\geq 6$ there is a non-empty range
$r+2\alpha+3\leq g\leq r+3\alpha-3$ in which the existence is yet to be
determined by some other (hopefully more systematic) treatment.

  * (ii)

In Proposition 7.2 (i.e. in the proof of Theorem 3.2 for the Claim (3.2.A)),
we impose numerical restrictions such as $r\geq\alpha+1$ and $g\geq
r+3\alpha-2$ so that Lemma 2.7 works. As a consequence, we used general
$k$-gonal curves with $k\geq 4$. However, if we consider the family of
trigonal curves with Maroni invariant $m$, it is possible to eliminate the
numerical restriction $r\geq\alpha+1$ and the ease the genus assumption $g\geq
r+3\alpha-2$ somehow. However, if we do this, we get very ample line bundle
only with particular degrees, say $d\equiv 2g-2(\mathrm{mod}~{}3)$. To be
precise, for $\alpha\geq 3$

$\mathcal{H}^{\mathcal{L}}_{2g-2-3(\alpha-1),g,r}\neq\emptyset,$

which however can be regarded a consequence of a classically known results on
trigonal curves; cf. [28, Lemma 1].

The existence and the irreducibility of
$\mathcal{H}^{\mathcal{L}}_{g+r-\alpha,g,r}$ with $r\geq\alpha+1$ for certain
value of the genus $g$ which is small with respect to $\alpha$ and $r$ – say
$r+2\alpha-1\leq g\leq r+3\alpha+1$ – can also be carried out in a similar
manner as were done in Proposition 4.1, Theorem 5.1, Theorem 6.1 and
Propositon 6.2. By analyzing extremal curves, nearly extremal curves or curves
lying on a surface of degree $r$ in $\mathbb{P}^{r}$ it is possible to
determine the irreducibility of $\mathcal{H}^{\mathcal{L}}_{g+r-\alpha,g,r}$
some triples $(d,g,r)$. The following example – which is partially
complementary to Proposition 7.5 (a$\dagger$) – is among one of those.

######  Example 7.8.

$\mathcal{H}^{\mathcal{L}}_{d,g,r}=\mathcal{H}^{\mathcal{L}}_{3\alpha+6,3\alpha+3,\alpha+3}\neq\emptyset$
and is irreducible if $\alpha\geq 7$ and is reducible if $\alpha=6$.

##  Appendix A Specialization

We provide a proof of Proposition 2.5 for which the author could not find an
explicit source of a proof in the literature.

######  Proposition A.1.

(Propostion 2.5) Smooth curves in $\mathbb{P}^{n+1}$, $n\geq 2$ lying on a
cone over a rational normal curve in a hyperplane $H\cong\mathbb{P}^{n}$ is a
specialization of curves lying on a rational normal surface scroll.

###### Proof.

The proof is based on the proofs of similar statements in [9] and [30] for the
case $n=2$ and curves on cubic surfaces in $\mathbb{P}^{3}$. One first shows
that there is a flat family whose special fibre is a cone with a general fibre
a rational normal surface scroll as follows.

Let $T$ be the affine line $\mathbb{A}^{1}=\mathrm{Spec~{}}\mathbb{C}(t)$ and
let $x,y$ be the homogeneous coordinates of $\mathbb{P}^{1}$. Consider the
following non-splitting exact sequence,

$0\rightarrow\mathcal{O}_{\mathbb{P}^{1}}\stackrel{{\scriptstyle[x,y^{n-1}]}}{{\longrightarrow}}\mathcal{O}_{\mathbb{P}^{1}}(1)\oplus\mathcal{O}_{\mathbb{P}^{1}}(n-1)\stackrel{{\scriptstyle[y^{n-1},-x]^{t}}}{{\longrightarrow}}\mathcal{O}_{\mathbb{P}^{1}}(n)\longrightarrow
0.$ (28)

Since the exact sequence (28) does not split, we let
$0\neq\mathbf{e}\in\textrm{Ext}^{1}(\mathcal{O}_{\mathbb{P}^{1}}(n),\mathcal{O}_{\mathbb{P}^{1}})$
be the corresponding extension class of the exact sequence. Let
$\mathbb{P}^{1}_{T}$ be the projective line over $T$. The class
$t\,\mathbf{e}\in\textrm{Ext}^{1}(\mathcal{O}_{\mathbb{P}^{1}_{T}}(n),\mathcal{O}_{\mathbb{P}^{1}_{T}})$
induces an extension $\mathcal{E}$ of $\mathcal{O}_{\mathbb{P}^{1}_{T}}(n)$ by
$\mathcal{O}_{\mathbb{P}^{1}_{T}}$ corresponding to the exact sequence

$0\longrightarrow\mathcal{O}_{\mathbb{P}^{1}_{T}}\longrightarrow\mathcal{E}\longrightarrow\mathcal{O}_{\mathbb{P}^{1}_{T}}(n)\longrightarrow
0.$

The fiber of the vector bundle $\mathcal{E}$ over $t\in T$ is given by

$\mathcal{E}_{t}\cong\mathcal{O}_{\mathbb{P}^{1}}(1)\oplus\mathcal{O}_{\mathbb{P}^{1}}(n-1)\textrm{
~{}for ~{}}t\in T-\\{0\\}$

and

$\mathcal{E}_{0}\cong\mathcal{O}_{\mathbb{P}^{1}}\oplus\mathcal{O}_{\mathbb{P}^{1}}(n)$

by definition; since
$t=0,t\,\mathbf{e}=\mathbf{0}\in\textrm{Ext}^{1}(\mathcal{O}_{\mathbb{P}^{1}_{T}}(n),\mathcal{O}_{\mathbb{P}^{1}_{T}})$
hence the corresponding exact sequence splits. Set

$Y=\mathbb{P}(\mathcal{E})\stackrel{{\scriptstyle\pi}}{{\rightarrow}}\mathbb{P}^{1}_{T},$

a $\mathbb{P}^{1}$ bundle over $\mathbb{P}^{1}_{T}$. Note that

$Y_{t}\cong\mathbb{P}(\mathcal{O}_{\mathbb{P}^{1}}(1)\oplus\mathcal{O}_{\mathbb{P}^{1}}(n-1))\cong\mathbb{F}_{n-2}\textrm{
for }t\in T-\\{0\\}$

and

$Y_{0}\cong\mathbb{P}(\mathcal{O}_{\mathbb{P}^{1}}\oplus\mathcal{O}_{\mathbb{P}^{1}}(n))\cong\mathbb{F}_{n}.$

We have the following diagram

$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi}$$\scriptstyle{\pi}$$\textstyle{~{}~{}~{}\mathbb{P}^{n+1}_{T}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p}$$\textstyle{\mathbb{P}^{1}_{T}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p}$$\textstyle{T}$

where $\varphi$ is the morphism induced by the tautological bundle
$\mathcal{L}$ on $\mathbb{P}(\mathcal{E})=Y$; the image $S$ under $\varphi$ is
a family of surfaces in $\mathbb{P}^{n+1}$ of degree $n$ over the affine line
$T$ whose general fiber is a rational normal scroll in $\mathbb{P}^{n+1}$ and
the special fiber is a cone over a rational normal curve in $\mathbb{P}^{n}$.
This family is a flat family because the Hilbert polynomial of the image of
special and general fibers are the same.

Let $X_{0}$ be a smooth connected curve of degree $d$ and genus $g$ on the
cone $S_{0}\subset\mathbb{P}^{n+1}$ over a rational normal curve in
$\mathbb{P}^{n}$. Let $\tilde{X_{0}}\subset Y_{0}$ denote the proper
transformation of $X_{0}$ under the desingularization $Y_{0}\rightarrow
S_{0}$. Denoting $h$ and $f$ by the class of the pull back of
$\mathcal{O}_{S_{0}}(1)$ via $Y_{0}\rightarrow S_{0}$ and the class of a fiber
$Y_{0}\rightarrow\mathbb{P}^{1}$, we have

$\tilde{X_{0}}\equiv
kh+(d-nk)f\in\mathrm{Pic}{~{}Y_{0}}\cong\mathbb{Z}^{\oplus 2},$

where $k:=(\tilde{X_{0}}\cdot f)$.

Set

$\mathcal{M}=\mathcal{L}^{\otimes
k}\otimes\pi^{*}\mathcal{O}_{\mathbb{P}^{1}_{T}}(d-nk).$

Then for any $t\in T$ the fiber $\mathcal{M}_{t}$ over $t$ is an inversitble
sheaf on $Y_{t}$, so that

$\mathcal{M}_{t}\equiv kh+(d-nk)f\in\mathrm{Pic}~{}Y_{t}$

by definition of $\mathcal{M}$; by abusing notation we use the same notaton
$f$ and $h$ on $\mathrm{Pic}~{}Y_{t}$ denoting the fiber and the pull back of
$\mathcal{O}_{S_{t}}(1)$ via
$Y_{t}\stackrel{{\scriptstyle\cong}}{{\rightarrow}}S_{t}$ which is an
isomorphism if $t\neq 0$. By (5), we have $d=nk$ or $d=nk+1$. We claim that
for every $t\neq 0$ there exists smooth connected curve $X_{t}\subset
S_{t}\subset\mathbb{P}^{n+1}$ such that $\tilde{X_{t}}\in|\mathcal{M}_{t}|$;
here we use the notation $\tilde{X_{t}}$ as a curve on the surface
$\mathbb{F}_{n-2}\cong Y_{t}$. To see this, note that the corresponding line
bundle on $\mathbb{F}_{n-2}$ is very ample; if $d=nk$, then
$\mathcal{M}_{t}\equiv kh$ and since $h=C_{0}+(n-1)f$ is very ample,
$\mathcal{M}_{t}$ is very ample. If $d=nk+1$, then

$\mathcal{M}_{t}\equiv kh+f=k(C_{0}+(n-1)f))+f=kC_{0}+((n-1)k+1)f$

is still very ample by the well-known criteria for the very ampleness of line
bundles on $\mathbb{F}_{n-2}\cong Y_{t}\cong S_{t}$ if $t\neq 0$; cf. [19, V.
Corollary 2.18]. $h^{0}(Y_{t},\mathcal{M}_{t})=g+2d-(n-2)k$ for $t\neq 0$ by a
formula corresponding to (1) on a smooth scroll in $\mathbb{P}^{n+1}$. For
$t=0$ we also have

$h^{0}(Y_{0},\mathcal{M}_{0})=g+2d-(n-2)k,$

coming from the vanishing of the cohomology on $\mathbb{F}_{e}$, $e\geq 0$;

$h^{i}(\mathbb{F}_{e},mh)=h^{i}(\mathbb{F}_{e},mh+f)=0\mathrm{~{}for~{}}i=1,2\mathrm{~{}and~{}}m\geq-1$
(29)

Recall that $Y_{0}=\mathbb{F}_{n}$ and $\mathcal{M}_{0}=kh+f$ or $kh$.

$h^{i}(Y_{0},\mathcal{M}_{0})=h^{i}(\mathbb{F}_{n},\tilde{X_{0}})=0$

for $i=1,2$ by (29). By Riemann-Roch,

$\displaystyle h^{0}(Y_{0},\mathcal{M}_{0})$ $\displaystyle=$ $\displaystyle
h^{0}(Y_{0},\tilde{X}_{0})=\frac{1}{2}(\tilde{X}_{0}+K_{Y_{0}})\cdot\tilde{X}_{0}-\tilde{X}_{0}\cdot
K_{Y_{0}}+\chi(\mathcal{O}_{Y_{0}})$ $\displaystyle=$
$\displaystyle\frac{1}{2}\deg
K_{\tilde{X}_{0}}-\tilde{X}_{0}\cdot(-2h+(n-2)f)+1$ $\displaystyle=$
$\displaystyle g+2d-(n-2)k.$

We may regard $Y$ as a $T$-schme via

$g:=\varphi\circ p=p\circ\pi;Y\rightarrow T.$

Note that by Grauert theorem $g_{*}\mathcal{M}$ is locally free of rank
$r=g+2d-(n-2)k$ on $T$ and hence $\mathbb{P}(g_{*}\mathcal{M})$ is an
irreducible $\mathbb{P}^{r-1}$ -bundle over $T$. We further that a general
elment of $\mathbb{P}(g_{*}\mathcal{M})$ corresponds to a smooth connected
curve which is conained in a general member of the family
$S\subset\mathbb{P}^{n+1}_{T}$. Therefore it finally follows that there is an
open set $U\subset\mathbb{P}(g_{*}\mathcal{M})$ and a flat family of curves
over $U$ whose general element is a curve on the surface
$S_{t}\subset\mathbb{P}^{n+1}$, $t\in T-\\{0\\}$, and whose special member is
in $X_{0}\subset S_{0}\subset\mathbb{P}^{n+1}$ where $S_{0}$ is a cone; take
an open set $0\in V\subset T$. Then take $\tilde{U}=g^{-1}(V)\subset Y$. And
then consider the restriction map
$\tilde{g}=g_{|\tilde{U}}:\tilde{U}\rightarrow V\subset T$. Then take
$U=\mathbb{P}(\tilde{g}_{*}(\mathcal{M}_{|\tilde{U}})).$ ∎

##  Appendix B A criteria for linear normality

The criteria we used for the linear normality of curves lying on certain
rational surfaces is an immediate consequence of the following lemma due to
Dolcetti and Pareschi [12, Lemma 1.3].

######  Lemma B.1.

Let $C\subset\mathbb{P}^{n}$ be a smooth, connected curve and $H$ be a
hyperplane intersecting $C$ transversally. Let $Z\subset H$ be a curve. Assume
that

(i) $h^{j}(H,\mathcal{I}_{Z,H}(1))=0,j=0,1$.

(ii) $C\cap Z=C\cap H$ as schemes.

Then $X=C\cup Z$ satisfies $h^{j}(\mathbb{P}^{n},\mathcal{I}_{X}(1))=0,j=0,1$.

######  Corollary B.2.

Let $S\subset\mathbb{P}^{n}$ be a smooth rational surface $\mathbb{P}^{2}_{s}$
embedded by a very ample linear system $\mathcal{Z}=(a;b_{1},\cdots,b_{s})$,
$b_{1}\geq b_{2}\geq\cdots\geq b_{s}\geq 1$ with $\sum b_{i}<3a$. Let
$\mathcal{L}=(\tilde{a};\tilde{b}_{1},\cdots,\tilde{b}_{s})$ be a very ample
linear system such that $\tilde{a}\geq a,\tilde{b}_{1}\geq
b_{1},\cdots,\tilde{b}_{s}\geq b_{s}$. Assume that the linear system
$\mathcal{L}\otimes\mathcal{Z}^{-1}$ contains a smooth connected curve
$C\subset S$. Then a general $\tilde{C}\in\mathcal{L}$ is smooth, irreducible
and linearly normal.

###### Proof.

Let $Z=S\cap H\in\mathcal{Z}$ be a general hyperplane section of $S$ which is
smooth, irreducible and non-degenerate in $H$. Note that

  * (i)

$\deg Z=\deg S=Z^{2}=a^{2}-\sum b_{i}^{2}$

  * (ii)

$p_{a}(Z)=\frac{(a-1)(a-2)}{2}-\sum\frac{b_{i}(b_{i}-1)}{2}$ by adjunction

  * (iii)

$h^{0}(H,\mathcal{I}_{Z,H}(1))=0$ since $Z$ is non-degenerate in $H$

  * (iv)

$h^{1}(H,\mathcal{I}_{Z,H}(1))=0$: By $\sum b_{i}<3a$, $2p_{a}(Z)-2<\deg Z$
and we have

$n=h^{0}(H,\mathcal{O}_{H}(1))=h^{0}(Z,\mathcal{O}_{Z}(1))=\deg Z-p_{a}(Z)+1.$

Thus $h^{1}(H,\mathcal{I}_{Z,H}(1))=0$

The effective divisor $X=C+Z\in\mathcal{L}$ satisfiles
$h^{j}(\mathbb{P}^{n},\mathcal{I}_{X}(1))=0,j=0,1$ by Lemma B.1. Hence by
semi-continuity, a general element of the very ample $\mathcal{L}$ is a
linearly normal, smooth and irreducible curve. ∎

######  Remark B.3.

(i) For $a=3,1\leq s\leq 8,b_{1}=\cdots=b_{s}=1$, a smooth curve on a del
Pezzo surface of degree $9-s$ in $\mathbb{P}^{9-s}$ belonging to a very ample
linear system $({a}^{\prime};{b_{1}}^{\prime},\cdots{b_{s}}^{\prime})$ is
linearly normal if $b^{\prime}_{s}\geq 1$. For $s=6$, this is a special case
of a stronger result by Kleppe [27, Proposition 3.1.3]

(ii) If $a=4,s=10,b_{i}=1,1\leq i\leq 10$, a smooth curve
$C\in(a^{\prime};b^{\prime}_{1},\cdots,b^{\prime}_{s}),b_{1}^{\prime}\geq
b^{\prime}_{2}\geq\cdots\geq b^{\prime}_{s}$ on a Bordiga surface
$S\stackrel{{\scriptstyle(4;1^{s})}}{{\hookrightarrow}}\mathbb{P}^{4}$ is
linearly normal if $b^{\prime}_{s}\geq 1$.

##  Appendix C Very ample linear series on a triple covering of elliptic
curves

######  Lemma C.1.

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

###### Proof.

$\mathcal{E}=|\eta^{*}(g^{\alpha-1}_{\alpha})|$ is base-point-free since
$g^{\alpha-1}_{\alpha}$ on $E$ is base-point-free. Set
$\mathcal{E}_{0}=\eta^{*}(g^{\alpha-1}_{\alpha})\subset|\eta^{*}(g^{\alpha-1}_{\alpha})|$.
Suppose $\dim\mathcal{E}=\beta\geq\alpha$. Note that the morphism induced by
$\mathcal{E}_{0}=g^{\alpha-1}_{3\alpha}$ which is same as the triple cover
$\eta$ factors through the morphism
$C\stackrel{{\scriptstyle\varphi}}{{\rightarrow}}\mathbb{P}^{\beta}$ induced
by $\mathcal{E}$ and $\deg\varphi|\deg\eta=3$. If $\deg\varphi=3$, then
$\alpha\leq\beta\leq\deg\varphi(C)=\frac{3\alpha}{\deg\varphi}=\alpha$ and
hence $\varphi(C)$ is a rational curve, which is impossible (by Castelnuovo-
Severi inequality or by the fact that there is no morphism from a rational
curve $\varphi(C)$ onto an elliptic curve $E$. Hence the morphism $\varphi$ is
birationally very ample. However $g\leq\pi(3\alpha,\alpha)=3\alpha+3$ which is
not compatible with the genus assumption $g\geq 3\alpha+7$. Thus we have
$\dim\mathcal{E}=\alpha-1$.

Suppose $\mathcal{E}^{\vee}=\mathcal{D}$ is not very ample. Choose $\Delta\in
C_{2}$ such that
$\mathcal{E}^{\prime}=|\mathcal{E}+\Delta|=g^{s}_{3\alpha+2}$, $\alpha\leq
s\leq\alpha+1$. Since $|\mathcal{E}^{\prime}-\Delta|=\mathcal{E}$, the triple
covering
$C\stackrel{{\scriptstyle\eta}}{{\rightarrow}}E\subset\mathbb{P}^{\alpha-1}$
onto $E$ factors through the morphism
$C\stackrel{{\scriptstyle\zeta}}{{\rightarrow}}F\subset\mathbb{P}^{s}$ induced
by the moving part $\widetilde{\mathcal{E}}$ of $\mathcal{E}^{\prime}$;

$\textstyle{C\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\zeta}}$$\scriptstyle{\eta}$$\textstyle{{F}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\nu}$$\textstyle{E}$

Note that

$\deg{\zeta}|\deg\eta=3,\,\deg F\geq
s\geq\alpha,\,3\alpha+1\leq\deg\widetilde{\mathcal{E}}=\deg\zeta\cdot\deg
F\leq 3\alpha+2.$

Hence $\deg{\zeta}=1$ and $\zeta$ is a birational morphism onto its image
$F\subset\mathbb{P}^{s}$. By the Castelnuovo genus bound,

$g\leq p_{a}(F)\leq\pi(t,s)~{}~{}~{}\textrm{with}~{}~{}~{}3\alpha+1\leq t\leq
3\alpha+2,\alpha\leq s\leq\alpha+1$

and we have the following cases;

$g\leq p_{a}(F)\leq\pi(t,s)=\begin{cases}3\alpha&\textrm{if
}~{}(t,s)=(3\alpha+1,\alpha+1)\\\ 3\alpha+3&\textrm{if
}~{}(t,s)=(3\alpha+2,\alpha+1)\\\ 3\alpha+6&\textrm{if
}~{}(t,s)=(3\alpha+1,\alpha),\alpha\geq 4\\\ 3\alpha+7&\textrm{if
}~{}(t,s)=(3\alpha+1,\alpha),\alpha=3\\\ 3\alpha+9&\textrm{if
}~{}(t,s)=(3\alpha+2,\alpha),\alpha\geq 5\\\ 3\alpha+10&\textrm{if
}~{}(t,s)=(3\alpha+2,\alpha),\alpha=4\\\ 3\alpha+11&\textrm{if
}~{}(t,s)=(3\alpha+2,\alpha),\alpha=3\par\par\end{cases}$

Three cases from the top can be eliminated by the assumption $g\geq
3\alpha+7$. For the fourth case $\alpha=3$, $(t,s)=(10,3)$, we have
$g=3\alpha+7=16=\pi(10,3)$. Hence $F\subset\mathbb{P}^{3}$ is an extremal
curve and is $5$-gonal. By the Casetelnuovo-Severi inequality, a curve of
genus $g\geq 12$ cannot be both triple covering of an elliptic curve and
$5$-gonal. For the remaining three cases, while we have $\pi(t,s)\gneq
3\alpha+7$, we also have

$\pi_{1}(3\alpha+2,\alpha)=3\alpha+6<g$

hence $F\subset\mathbb{P}^{\alpha}$ lies on a surface
$S\subset\mathbb{P}^{\alpha}$ of minimal degree $\alpha-1$. We assume $S$ is a
smooth rational normal scroll or a Veronese if $\alpha=5$. If $\alpha=5$,
$\deg F=t=3\alpha+2$ is odd and hence $S$ is cannot be a Veronese surface.
Using the same notation as in Remark 2.3, we let $F\in|aH+bL|$. We find common
integer solutions $(a,b)$ for the equations (2) and (3) in Remark 2.3 by
substituting $n=\alpha-1$, $d=t=3\alpha+2$,
$p_{a}(F)=3\alpha+7,3\alpha+8,3\alpha+9$, $3\alpha+10$ (only when
$\alpha=3,4)$ and $3\alpha+11$ (only when $\alpha=3$). After elementary but
tedious numerical computation, we arrive at

$(a,b)=\begin{cases}(6,-1)&\textrm{if }~{}p_{a}(F)=3\alpha+11;\textrm{ only
~{}when }\alpha=3\\\ (5,-1)&\textrm{if }~{}p_{a}(F)=3\alpha+10;\textrm{ only
~{}when }\alpha=4\\\ (4,6-\alpha)&\textrm{if }~{}p_{a}(F)=3\alpha+9\textrm{ in
general\quad or }\\\ (5,-3)&\textrm{if }~{}p_{a}(F)=3\alpha+9\textrm{ and
}\alpha=5\\\ (7,-3)&\textrm{if }~{}p_{a}(F)=3\alpha+9\textrm{ and }\alpha=3\\\
(5,-5)&\textrm{if }~{}p_{a}(F)=3\alpha+8;\textrm{ only ~{}when }\alpha=6\\\
(6,-4)&\textrm{if }~{}p_{a}(F)=3\alpha+8;\textrm{ only ~{}when }\alpha=4\\\
(5,-7)&\textrm{if }~{}p_{a}(F)=3\alpha+7;\textrm{ only ~{}when }\alpha=7.\\\
\end{cases}$

Note that the ruling $|L|$ cut out a base-point-free $g^{1}_{a}$ on
$F\in|aH+bL|$. Hence by the Castelnuovo-Severi inequality, we have $g\leq
2a+1\leq 15\leq 3\alpha+6$ if $a\leq 7$ and $\alpha\geq 3$, contrary to the
genus assumption $g\geq 3\alpha+7$.

We assume that $S\subset\mathbb{P}^{\alpha}$ is a cone over a rational normal
curve in $\mathbb{P}^{\alpha-1}$. In this case we solve the equation (4) in
Remark 2.3 to find integer values $k$ by substituting $n=\alpha-1$,
$d=3\alpha+2$. $p_{a}(F)=3\alpha+7,3\alpha+8,3\alpha+9$, $3\alpha+10$ (only if
$\alpha=3,4)$ and $3\alpha+11$ (only if $\alpha=3$);

$k=\begin{cases}5,6&\textrm{if }~{}p_{a}(F)=3\alpha+11;\textrm{ only ~{}when
}\alpha=3\\\ 5&\textrm{if }~{}p_{a}(F)=3\alpha+10;\textrm{ only ~{}when
}\alpha=4\\\ 4&\textrm{if }~{}p_{a}(F)=3\alpha+9\textrm{ in general or }\\\
5&\textrm{if }~{}p_{a}(F)=3\alpha+9\textrm{ only when }\alpha=5\\\
7&\textrm{if }~{}p_{a}(F)=3\alpha+9\textrm{ only when }\alpha=3\\\
5&\textrm{if }~{}p_{a}(F)=3\alpha+8;\textrm{ only ~{}when }\alpha=6\\\
5&\textrm{if }~{}p_{a}(F)=3\alpha+7;\textrm{ only ~{}when }\alpha=7.\\\
\end{cases}$

Recall that $k=\widetilde{F}\cdot f$ on the Hirzebruch surface
$\widetilde{S}\cong\mathbb{F}_{\alpha-1}\rightarrow S$, where $\widetilde{F}$
is the strict transformation of the reduced curve $F\subset
S\subset\mathbb{P}^{\alpha}$ under the desingularization
$\widetilde{S}\cong\mathbb{F}_{\alpha-1}\rightarrow S$ of the cone $S$; cf.
Remark 2.3 (iii). Hence $C\cong\widetilde{F}$ has $g^{1}_{k}$ and we arrive at
the same contradiction as the smooth scroll case by Castelnuovo-Severi
inequality. ∎

## References

  * [1] R. Accola, Topics in the theory of Riemann surfaces. Lecture Notes in Mathematics 1595, Springer, Heidelberg, 1991.
  * [2] E. Arbarello and M. Cornalba, A few remarks about the variety of irreducible plane curves of given degree and genus. Ann. Sci. Éc. Norm. Supér. (4) 16 (1983), 467–483.
  * [3] E. Arbarello, M. Cornalba, P. Griffiths and J. Harris, Geometry of Algebraic Curves Vol.I. Springer-Verlag, Berlin/Heidelberg/New York/Tokyo, 1985.
  * [4] E. Arbarello, M. Cornalba and P. Griffiths, Geometry of Algebraic Curves Vol.II. Springer, Heidelberg, 2011.
  * [5] E. Ballico, C. Fontanari and C. Keem, On the Hilbert scheme of linearly normal curves in $\mathbb{P}^{r}$ of relatively high degree. J. Pure Appl. Algebra (2020) 224 (2020), 1115–1123.
  * [6] E. Ballico, C. Keem, On the Hilbert scheme of smooth curves of degree $15$ in $\mathbb{P}^{5}$. Preprint, available at http://arxiv.org/abs/2310.00682.
  * [7] A. Beauville, Complex algebraic surfaces. Cambridge University Press, London/New York, 1983.
  * [8] J. Blanc, Finite subgroups of the Cremona group of the plane preprint, available at mimuw.edu.pl/ jarekw/EAGER/pdf/FiniteSubgroupsCremona.pdf .
  * [9] J. Brevik, Curves on normal rational cubic surfaces, Pacific J. Math. 230 (2007), no. 1, 73–105.
  * [10] M. Coppens, C. Keem and G. Martens, The primitive length of a general $k$-gonal curve. Indag. Math. (N.S.), 5 (1994), no. 2, 145–159.
  * [11] T. Dedieu and E. Sernesi, Equigeneric and equisingular families of curves on surfaces. Publ. Mat. , 61 (2017), 175—212.
  * [12] A. Dolcetti and G. Pareschi, On linearly normal space curves. Math. Z., 198 (1988), 73–82.
  * [13] S. Di Rocco, $k$-very ample line bundles on Del-Pezzo surfaces. Math. Nachr., 179 (1996), 47–56.
  * [14] L. Ein, Hilbert scheme of smooth space curves. Ann. Scient. Ec. Norm. Sup. (4), 19 (1986), no. 4, 469–478.
  * [15] L. Gruson and C. Peskine, Genre des courbes de l’espace projectif II. Ann. Scient. Ec. Norm. Sup. (4), 15 (1982), no. 3, 401–418.
  * [16] J. Harris, Curves in Projective space. in “Sem. Math. Sup.,”, Press Univ. Montréal, Montréal, 1982.
  * [17] J. Harris, On the Severi problem. Invent. Math 84, (1986), 445–461.
  * [18] J. Harris and I. Morrison, Moduli of curves. Springer-Verlag, Berlin/Heidelberg/New York, 1991.
  * [19] R. Hartshorne, Algebraic Geometry. Springer-Verlag, Berlin/Heidelberg/New York, 1977.
  * [20] V. Kanev, Hurwitz spaces of triple coverings of elliptic curves and moduli spaces of Abelian threefolds. Ann. Mat. Pura Appl. IV. Ser. 183 (2004), 333?374.
  * [21] C. Keem, Reducible Hilbert scheme of smooth curves with positive Brill-Noether number. Proc. Amer. Math. Soc., 122 (1994), no. 2, 349–354.
  * [22] C. Keem, Existence and the reducibility of the Hilbert scheme of linearly normal curves in $\mathbb{P}^{r}$ of relatively high degrees, J. Pure Appl. Algebra 227 (2023), 1115–1123, available at https://arxiv.org/abs/2101.00559.
  * [23] C. Keem, On the Hilbert scheme of linearly normal curves in $\mathbb{P}^{r}$ with small index of speciality. Indag. Math.(N.S.), 33 (2022), 1102–1124, available at https://arxiv.org/abs/2103.16321.
  * [24] C. Keem, On the Hilbert scheme of linearly normal curves in $\mathbb{P}^{r}$ with higher index of speciality. In preparation.
  * [25] C. Keem and Y.-H. Kim,On the Hilbert scheme of linearly normal curves in $\mathbb{P}^{4}$ of degree $d=g+1$ and genus $g$. Arch. Math., 113 (2019), no. 4, 373–384.
  * [26] C. Keem, Y.-H. Kim and A.F. Lopez, Irreducibility and components rigid in moduli of the Hilbert Scheme of smooth curves. Math. Z., 292 (2019), no. 3-4, 1207–1222.
  * [27] J. Kleppe, The Hilbert-flag scheme, its properties and its connection with the Hilbert scheme. Thesis, Oslo, 1982.
  * [28] G. Martens and F.-O. Schreyer, Line bundles and sygygies of trigonal curves, Abh. Math. Sem. Univ. Hamburg 56 (1985), 169–189.
  * [29] S. Mori, On degrees and genera of curves on smooth quartic surface in $\mathbb{P}^{3}$. Nagoay Math. J.,
  * [30] H. Nasu, The Hilbert scheme of space curves of degree $d$ and genus $3d-18$. Comm. Algebra 36 (2008), 4163-4185.
  * [31] M. Ohkouchi and F. Sakai The gonality of singular plane curves. Tokyo J. Math., 27 (2004), 137-147.
  * [32] J. Rathmann The genus of curves in $\mathbb{P}^{4}$ and $\mathbb{P}^{5}$. Math. Z., 202 (1989), 525-543.
  * [33] F. Severi, Vorlesungen über algebraische Geometrie.Teubner, Leipzig, 1921.
  * [34] I. Tyomkin, On Severi varieties on Hirzebruch surfaces Int. Math. Res. Not., 23 (2007), Art. ID rnm109.