A bound on the genus of a curve with Cartier operator of small rank

Abstract

Ekedahl showed that the genus of a curve in characteristic $p>0$ with zero Cartier operator is bounded by $p(p-1)/2$. We show the bound $p+p(p-1)/2$ in case the rank of the Cartier operator is 1, improving a result of Re.

1 Introduction

In [1] Ekedahl gave a bound for the genus g of an irreducible smooth complete curve over an algebraically closed field of characteristic $p>0$ with zero Cartier operator: $g\le p(p-1)/2$. This bound is sharp and was generalized by Re to curves with Cartier operator of given rank [2]. He showed for hyperelliptic curves whose Cartier operator has rank m the bound $g<mp+(p+1)/2$, and for non-hyperelliptic curves

$$\begin{aligned} g\le mp+(m+1)p(p-1)/2. \end{aligned}$$

(1)

He also showed that if the Cartier operator ${\mathcal {C}}$ is nilpotent and ${\mathcal {C}}^r=0$, then

$$\begin{aligned} g\le p^r(p^r-1)/2. \end{aligned}$$

In this paper we give a strengthening of the result (1) of Re. One can find other related results in [3, 4] and [5].

Theorem 1.1

Let X be an irreducible smooth complete curve of genus g over an algebraically closed field of characteristic $p>0$. If the rank of the Cartier operator of X equals 1, we have $g\le p(p+1)/2$.

This is sharp for example for $p=2$, see [6, Lemma 4.8]. In the case of higher rank we have the following result.

Theorem 1.2

Let X be an irreducible smooth complete curve of genus g over an algebraically closed field of characteristic $p>0$. If the rank of the Cartier operator of X equals 2, and if X possesses a point R such that linear system |pR| is base point free, then $g\le p(p+3)/2$, while if X does not have such a point, one has the bound $g\le p(4p+1)/3$.

2 The Cartier operator and linear systems

From now on, by a curve we mean an irreducible smooth complete curve over an algebraically closed field k of characteristic $p>0$. For a curve X with function field k(X), Cartier [7] defined an operator on rational differential forms with the following properties:

• (1) ${\mathcal {C}}(\omega _1+\omega _2)={\mathcal {C}}(\omega _1)+{\mathcal {C}}(\omega _2),$

• (2) ${\mathcal {C}}(f ^p\omega )=f\, {\mathcal {C}}(\omega ),$

• (3) ${\mathcal {C}}(\mathop {}\!\mathrm {d}f)=0,$

• (4) ${\mathcal {C}}(\mathop {}\!\mathrm {d}f/f)=\mathop {}\!\mathrm {d}f/f,$

where $f \in k(X)$ is non-zero. Moreover, recall that if x is a separating variable of k(X), any $f\in k(X)$ can be written as

$$\begin{aligned} f=f_0^p+\cdots +f^p_{p-1}x^{p-1},~\text {with}~f_i\in k(X). \end{aligned}$$

(2)

For a rational differential form $\omega = f\mathop {}\!\mathrm {d}x$ with f as in (2), we have ${\mathcal {C}}(\omega )=f_{p-1}\mathop {}\!\mathrm {d}x$. In particular ${\mathcal {C}}^n(f^i\mathop {}\!\mathrm {d}f)=f^{(i+1)/p^n-1}\mathop {}\!\mathrm {d}f$ if $p^n|i+1$, and ${\mathcal {C}}^n(f^i\mathop {}\!\mathrm {d}f)=0$ otherwise. Furthermore, for distinct points $Q_1$, $Q_2$ on X, if there is a rational differential form $\omega$ that ${\mathop {}\!\mathrm {ord}_{Q_1}(\omega )\ge p}$ and $\mathop {}\!\mathrm {ord}_{Q_2} (\omega )=p-1$, then by property 2) above we have $\mathop {}\!\mathrm {ord}_{Q_1}({\mathcal {C}}(\omega ))\ge 1$ and $\mathop {}\!\mathrm {ord}_{Q_2}({\mathcal {C}}(\omega ))=0$.

This operator ${\mathcal {C}}$ induces a map ${\mathcal {C}}: \, H^0(X,\varOmega _X^1) \rightarrow H^0(X,\varOmega _X^1)$ which is $\sigma ^{-1}$-linear, that is, it satisfies properties (1) and (2) above, with $\sigma$ denoting the Frobenius automorphism of k. We are interested in the relation between the rank of the Cartier operator, defined as $\dim _k{\mathcal {C}}(H^0(X,\varOmega _X^1))$, and the genus g.

Re showed that there is a relation between the rank of Cartier operator and the geometry of linear systems on a curve. We will list some results that we will use and refer for the proof to Re’s paper [2]. In the following, X denotes a non-hyperelliptic curve and for D a divisor on X, we will denote by $H^i(D)$ the vector space $H^i(X,{\mathcal {O}} _X (D))$.

We will say that a statement holds for a general effective divisor of degree n on X if the statement is true for divisors in a nonempty open set of effective divisors of degree n on X. By a general point we mean a general effective divisor of degree 1. We start with a few results of Re.

Proposition 2.1

[2, Prop. 2.2.2] Let X be a non-hyperelliptic curve with $\mathrm{rank}({\mathcal {C}})=m$. Then for a general effective divisor $D=Q_1+\cdots +Q_{m+1}$ on X with $\deg D=m+1$, one has

$$\begin{aligned} h^0(pD)=1+h^0(pD-Q_{m+1}). \end{aligned}$$

This implies for a general divisor D with $\deg D>\mathrm{rank}({\mathcal {C}})$, that the linear system |pD| is base point free. As a corollary, we have the following.

Corollary 2.2

[2, Prop. 2.2.3] If X is a non-hyperelliptic curve with zero Cartier operator, then $h^0(p\, Q)\ge 2$ for any point Q on X.

The following lemma gives a way of estimating dimensions of linear systems.

Lemma 2.3

[2, Lemma 2.3.1] Assume that $Q_1$ and $Q_2$ are general points on a non-hyperelliptic curve X and that D is a divisor. Then we have

$$\begin{aligned} h^0(pD+p\,Q_1+p\,Q_2)-h^0(pD+p\,Q_1)\ge h^0(pD+p\,Q_1)-h^0(pD). \end{aligned}$$

We now give a generalization of a result of Re.

Proposition 2.4

Let D,E,F be effective divisors on a non-hyperelliptic curve X such that

(1):

|F| is base point free;

(2):

$D > 0$ and $\mathrm{Supp}(D) \cap \mathrm{Supp}(E)=\emptyset$;

(3):

There are points $Q_1,\ldots ,Q_{m+1} \in \mathrm{Supp}(D)$ and a divisor $F_1 \in |F|$ such that $\mathop {}\!\mathrm {ord}_{Q_i}(F_1)=1$ for $1\le i \le m+1$ and $\mathrm{Supp}(D)\cap \mathrm{Supp}(F_1)=~\{Q_1,\ldots ,Q_{m+1}\}$;

(4):

For these points $Q_i$ one has $h^0(E+\sum \nolimits _{i=1}^{m+1}Q_i)=h^0(E)$;

(5):

$Q_i$ is not a base point of $|D+E+F|$ for $i =1,\ldots ,m+1$ and there exist $s_1,\ldots ,s_{m+1}\in H^0(D+E+F)$ such that

$$\begin{aligned} \mathop {}\!\mathrm {ord}_{Q_i}(s_i)=0, ~ \mathop {}\!\mathrm {ord}_{Q_i}(s_j)\ge p, ~~i\ne j,~i,j=1,\ldots ,m+1. \end{aligned}$$

Then we have

$$\begin{aligned} h^0(D+E+F)-h^0(E+F)\ge h^0(D+E)-h^0(E)+m+1. \end{aligned}$$

Proof

Let $s_{F_1}\in H^0(F)$ with divisor $F_1$ and $s_D\in H^0(D)$ with divisor D. We have a commutative diagram with exact rows:

Claim 2.5

Multiplication by $s_{F_1}$ induces an injective map

This Claim follows if

$$\begin{aligned} s_{F_1}\cdot H^0(D+E) \cap s_D\cdot H^0(E+F)=s_{F_1}\cdot s_D\cdot H^0(E). \end{aligned}$$

Because of assumptions (2) and (3), the left hand side of this equation is equal to $s_D\cdot s_{F_1}'\cdot H^0(E+\sum _{i=1}^{m+1}Q_i)$, where $s_{F_1}'=s_{F_1}/s_0$ for a section $s_0 \in H^0(\sum _{i=1}^{m+1}Q_i)$ with $\mathrm{div}(s_0)=\sum _{i=1}^{m+1}Q_i$. Then (4) implies $H^0(E+\sum _{i=1}^{m+1}Q_i)=s_0 \cdot H^0(E)$. The Claim follows.

By (5), there exist $s_1,\ldots , s_{m+1}$ such that for all i, j with $i\ne j$ we have $\mathop {}\!\mathrm {ord}_{Q_i}(s_i)=0$ and $\mathop {}\!\mathrm {ord}_{Q_i}(s_j)\ge p$. Now we will show that $s_1,\ldots ,s_{m+1}$ generate an $m+1$-dimensional subspace of $H^0(D+E+F)/s_D \cdot H^0(E+F)$ with zero intersection with $\mathop {}\!\mathrm {Im}(s_{F_1})$. First we will prove the zero intersection part. Assume there exist $c_1,\ldots , c_{m+1}\in k$ such that $\xi =\sum _{i=1}^{m+1}c_is_i$ lies in $\mathop {}\!\mathrm {Im}(s_{F_1})$. That means $\xi =s_{F_1}\cdot r+s_D\cdot t$ with some $r\in H^0(D+E)$ and $t\in H^0(E+F)$. If $c_1\ne 0$ then we obtain $\mathop {}\!\mathrm {ord}_{Q_1}(\xi )=0$. However, because $\mathop {}\!\mathrm {ord}_{Q_1}(F_1)=\mathop {}\!\mathrm {ord}_{Q_1}(s_{F_1})=1$ and $\mathop {}\!\mathrm {ord}_{Q_1}(s_D)=\mathop {}\!\mathrm {ord}_{Q_1}(D)\ge 1$, we have $0=\mathop {}\!\mathrm {ord}_{Q_1}(\xi )=\mathop {}\!\mathrm {ord}_{Q_1}(s_{F_1}\cdot r+s_D\cdot t)\ge 1$, a contradiction if $c_1\ne 0$. Similarly, we can show $c_2=\cdots =c_{m+1}=0$. Then for any non-zero element $\xi$ in $<s_1,\ldots ,s_{m+1}>$ one has $\xi \notin \mathop {}\!\mathrm {Im}(s_{F_1})$.

Now for the linear independence of $s_1,\ldots ,s_{m+1}$, if $\xi =\sum _{i=1}^{m+1}{c_is_i}$ lies in $s_D\cdot H^0(E+F)$, then $\xi =s_D\cdot t$ with $t \in H^0(E+F)$ and we can apply the same argument on the orders of $\xi$ at $Q_i$ as above with $r=0$. Then we find $c_i=0$ for $i=1,\ldots ,m+1$. So $s_1,\ldots ,s_{m+1}$ are linearly independent in $H^0(D+E+F)/s_D\cdot H^0(E+F)$.

By the injectivity Claim 2.5 above we then have

$$\begin{aligned} h^0(D+E+F)-h^0(D+E)\ge h^0(E+F)-h^0(E)+m+1. \end{aligned}$$

$\square$

3 Proofs of the Theorems 1.1 and 1.2

Before giving the proofs of theorems, we need several lemmas on the relation between the rank of the Cartier operator and geometrical properties of linear systems on a curve.

Lemma 3.1

Let X be a non-hyperelliptic curve with$~{\mathrm{rank}({\mathcal {C}})=m\ge 1}$. Then there exists points $Q_1, \ldots , Q_m$ on X such that with $D=\sum _{i=1}^mQ_i$ we have

$$\begin{aligned} h^0(pD)=1+h^0(pD-Q_m). \end{aligned}$$

Proof

Suppose that $\omega _1,\ldots ,\omega _m$ are differentials that generate $\mathop {}\!\mathrm {Im}({\mathcal {C}})$. Assume the lemma is not true, that is, for any m-tuple $\alpha =(Q_1,\ldots ,Q_m)$, we have with $D=\sum _{i=1}^{m}Q_i$ that $h^0(p\, D)=h^0(p\,D-Q_m)$. Then by Serre duality and Riemann–Roch, there exists a $\omega _{D} \in H^0(X,\varOmega _X^1)$ that

$$\begin{aligned} \mathop {}\!\mathrm {ord}_{Q_i}(\omega _{D}) \ge p, ~1\le i\le m-1,~ \mathop {}\!\mathrm {ord}_{Q_m}(\omega _{D}) =p-1. \end{aligned}$$

(3)

Let $\eta := {\mathcal {C}}(\omega _{D})=\sum _{i=1}^m{\lambda _i \, \omega _i}$ with $\lambda _i\in k$. Then one has

$$\begin{aligned} \mathop {}\!\mathrm {ord}_{Q_i}(\eta )&\ge 1, ~1\le i \le m-1, ~ \mathop {}\!\mathrm {ord}_{Q_m}(\eta ) =0. \end{aligned}$$

(4)

Suppose now that $\omega _1,\ldots ,\omega _m$ have a common base point R. Then define $Q_m=R$ and choose general points $Q_1,\ldots ,Q_{m-1}$ such that $Q_1,\ldots ,Q_{m-1},R$ form m distinct points. Then with $D=\sum _{i=1}^{m-1}Q_i+R$ we have $h^0(pD)=h^0(pD-R)$, hence there exists a $\omega _D$ satisfying (3). Then $\eta ={\mathcal {C}}(\omega _D)$ satisfies (4) and we have $0=\mathop {}\!\mathrm {ord}_{Q_m}(\eta )=\mathop {}\!\mathrm {ord}_{Q_m}(\sum _{i=1}^m \lambda _i\omega _i)\ge 1$, a contradiction.

So we may assume that $\omega _1,\ldots ,\omega _m$ have no common base point. Choose a point $Q_1$ such that $\omega _1$ does not vanish at $Q_1$, but $\omega _2,\ldots ,\omega _m$ vanish at $Q_1$. More generally, assume furthermore that we have $Q_1,\ldots ,Q_n$ such that $\mathop {}\!\mathrm {ord}_{Q_i}(\omega _i)=0$ and $\mathop {}\!\mathrm {ord}_{Q_i}(\omega _j)>0$ for $i=1,\ldots ,n$ and $i<j\le m$.

If $\omega _{n+1},\ldots ,\omega _m$ have a base point R different from $Q_i$ for $i=1,\ldots ,n$, then we choose $Q_{n+1},\ldots ,Q_{m-1}$ general distinct points, $Q_m=R$ and let $\alpha =(Q_1,\ldots ,Q_m)$. By assumption $h^0(p\, D)=h^0(p\, D-Q_m)$ for $D=\sum _{i=1}^mQ_i$, and we find a differential form $\omega _D$ satisfying (3) and therefore a form $\eta ={\mathcal {C}}(\omega _D)$ satisfying (4), again a contradiction.

So we may assume that $\omega _{n+1},\ldots ,\omega _m$ do not have common base points except $Q_1,\ldots ,Q_n$. Choose now a point $Q_{n+1}$ different from $Q_1,\ldots ,Q_n$ such that $\omega _{n+1}$ does not vanish at $Q_{n+1}$, but $\omega _{n+2},\ldots ,\omega _m$ all vanish at $Q_{n+1}$. By induction on n, we find points $Q_1,\ldots ,Q_{m-1}$ with $\mathop {}\!\mathrm {ord}_{Q_i}(\omega _i)=0$ and $\mathop {}\!\mathrm {ord}_{Q_i}(\omega _j)\ge 1$ for $j>i$ and $j=2,\ldots ,m$.

Now if $\omega _m$ has a zero distinct from $Q_i$ for $i=1,\ldots ,m-1$, say $Q_m$, we let $\alpha =(Q_1,\ldots ,Q_m)$ and $D=\sum _{i=1}^mQ_i$. The assumption $h^0(pD)=h^0(pD-Q_m)$ gives us a differential form $\omega _D$ and $\eta ={\mathcal {C}}(\omega _{D})=\sum _{i=1}^m \lambda _i \, \omega _i$. By (4) we have $0= \mathop {}\!\mathrm {ord}_{Q_m}(\eta )=\mathop {}\!\mathrm {ord}_Q(\lambda _m \omega _m)\ge 1$, a contradiction. So $\omega _m$ has no zeros outside $Q_1,\ldots ,Q_{m-1}$.

Now $\deg (\omega _m)=2g-2\ge m$ for $g\ge 2$, so $\omega _m$ vanishes at one $Q_i$ with multiplicity larger than one, say $Q_{m-1}$. Then with $D=\sum _{i=1}^{m-2}Q_i+2Q_{m-1}$ we have $h^0(pD)=h^0(pD-Q_{m-1})$, giving us a differential form $\omega _D$, and $\eta ={\mathcal {C}}(\omega _D)=\sum _{i=1}^m\lambda _i\, \omega _i$. Then we have $\mathop {}\!\mathrm {ord}_{Q_i}(\eta )\ge 1$ for $i=1,\ldots ,m-2$ and $\mathop {}\!\mathrm {ord}_{Q_{m-1}}(\eta )=1$. However, by the induction assumption

$$\begin{aligned} \mathop {}\!\mathrm {ord}_{Q_i}(\omega _i)&=0, \mathop {}\!\mathrm {ord}_{Q_i}(\omega _j)\ge 1,\quad 1\le i<j\le m-1,\\ \mathop {}\!\mathrm {ord}_{Q_l}(\omega _m)&\ge 1, \mathop {}\!\mathrm {ord}_{Q_{m-1}}(\omega _m)\ge 2,\quad l=1,2,\ldots ,m-2. \end{aligned}$$

So we must have $\lambda _i=0$ for $i=1,\ldots ,m-1$ and $\mathop {}\!\mathrm {ord}_{Q_{m-1}}(\eta )\ge 2$, and we therefore find $h^0(pD)=1+h^0(pD-Q_m)$. $\square$

By putting $m=1$ in the Lemma 3.1 above, we have the following.

Corollary 3.2

Let X be a non-hyperelliptic curve. If the Cartier operator has $\mathrm{rank}({\mathcal {C}})=1$, there exists a point R of X such that

$$\begin{aligned} h^0(pR)=1+h^0((p-1)R). \end{aligned}$$

Combining Lemma 3.1 above and Proposition 2.4, we have the following result. We denote the canonical divisor (class) by $K_X$.

Corollary 3.3

Let X be a non-hyperelliptic curve with $\mathrm{rank}({\mathcal {C}})=1$ and let $T_n$ be a general effective divisor of degree n. Put $E=p\, T_n$ and let R be a point of X with $h^0(pR)=1+h^0((p-1)R)$. Then the following holds.

$~\mathrm{(i)}$ If $h^0(K_X-E)\le 1$, one has for general points $Q_1$, $Q_2$

$$\begin{aligned} h^0\left( E+pR+\sum _{i=1}^2p\,Q_i\right) -h^0\left( E+\sum _{i=1}^2p\,Q_i\right) =p. \end{aligned}$$

$\mathrm{(ii)}$ If $h^0(K_X-E)\ge 2$, one has for general points $Q_1$, $Q_2$

$$\begin{aligned} h^0\left( E+pR+\sum _{i=1}^2p\,Q_i\right) -h^0\left( E+\sum _{i=1}^2p\,Q_i\right) \ge 2+h^0(E+pR)-h^0(E). \end{aligned}$$

Proof

Note that the existence of R is provided by Corollary 3.2 above.

• (i) If $h^0(K_X-E)=0$, i.e. E is non-special, Riemann–Roch implies statement i). If $h^0(K_X-E)=1$, we choose $Q_1$ a non-base point of $|K_X-E|$, then $h^0(K_X-E-Q_1)=0$, hence $h^0(K_X-E-p\,Q_1)=0.$ Therefore $h^0(K_X-E-\sum _{i=1}^2 p\,Q_i)=h^0(K_X-E-\sum _{i=1}^2 p\,Q_i-p\,R)=0$ and by Riemann–Roch we have $h^0(E+\sum _{i=1}^2 p\,Q_i+p\,R)-h^0(E+\sum _{i=1}^2 p\,Q_i)=p$.

• (ii) If $h^0(K_X-E)\ge 2$, we write $D=p\, Q_1+p\, Q_2 ,~ E=p\, T_n$ and $F=p\,R$ and we proceed to verify the conditions $(1){-}(5)$ of Proposition 2.4 in this case. Conditions (1) and (2) are easy consequences of the generality assumptions of $Q_1$, $Q_2$ and R. For condition (3), if the linear system |pR| induces a separable map to projective space, then we can choose $Q_1$ and $Q_2$ to be points where the map is smooth and find an effective divisor $F_1$ such that $\mathop {}\!\mathrm {ord}_{Q_1}(F_1)=\mathop {}\!\mathrm {ord}_{Q_2}(F_1)=1$. If, on the contrary, the map induced by |pR| is inseparable, then $\dim |R| \ge 1$, which is not true for curves of genus larger than zero.

Condition (4) is satisfied once we choose $Q_1$ to be a non-base point of $|K_X-E|$ and $Q_2$ a non-base point of $|K_X-E-Q_1|$, since $h^0(K_X-E)\ge 2$. Then we have $h^0(K_X-E-\sum _{i=1}^2Q_i)=h^0(K_X-E)-2$.

Condition (5) holds as $|E+pR+pQ_1+pQ_2|$ is base point free by Proposition 2.1 if $Q_1$ and $Q_2$ are general. Furthermore by Proposition 2.1, we have $h^0(E+pQ_i)=1+h^0(E+(p-1)Q_i)$ for $i=1,2$. Then we obtain $s_1$ and $s_2$ in $H^0(E+pR+pQ_1+pQ_2)=H^0(D+E+F)$ such that for all i, j we have $\mathop {}\!\mathrm {ord}_{Q_i}(s_i)=0$ and $\mathop {}\!\mathrm {ord}_{Q_i}(s_j)\ge p$ for $j\ne i$.

Then we conclude by Proposition 2.4 above. $\square$

Now we can state some numerical consequences of Corollary 3.3.

Corollary 3.4

Let X be a non-hyperelliptic curve with $\mathrm{rank}({\mathcal {C}})=1$. Denote by $D_n$ a general divisor of degree n. Then for any integer $n\ge 1$, one has

• (i)  $p\ge h^0(pD_{2n})-h^0(pD_{2n-1})\ge \min (2n-1,p)$.

• (ii)  For $1\le n \le \left\lceil (p+1)/2\right\rceil$, one has $p\ge h^0(pD_{2n-1})-h^0(pD_{2n-2})\ge ~{2n-2}$.

• (iii)   $pD_p$ is non-special, i.e. $h^0(K_X-pD_p)=0$.

• (iv)  For $1\le n \le \left[ (p+1)/2\right]$, one has $h^0(pD_{2n})-h^0(pD_{2n-2})\ge 4n-3$.

• (v)  For $1\le n \le \left[ (p+1)/2\right]$, one has

  $$\begin{aligned} h^0(K_X-pD_{2n-2})-h^0(K_X-pD_{2n})\le 2p-4n+3. \end{aligned}$$

• (vi)  $h^0(K_X-pD_{p-1})\le 1$ for $p\ge 3$.

Proof

(i) For $n\in {\mathbb {Z}}_{>0}$, one can always has$~p\ge h^0(pD_{2n})-h^0(pD_{2n-1})$. We will prove the second inequality in (i) by induction on n.

In the case $n=1$, by Proposition 2.1, for general points $Q_1$, $Q_2$ one has

$$\begin{aligned} h^0(pQ_1+pQ_2)=1+h^0(pQ_1+(p-1)Q_2), \end{aligned}$$

and thus with $D_2=Q_1+Q_2$ and $D_1=Q_1$, we see $h^0(pD_2)\ge 1+h^0(pD_1)$. Now we do induction and assume $h^0(pD_{2n-2})-h^0(pD_{2n-3})\ge 2n-3$. We apply Corollary 3.3 with $E=pD_{2n-3}$ for $n\ge 2$. If $h^0(K_X-E)\le 2$, then we have $h^0(pD_{2n})-h^0(pD_{2n-1})=p$. Otherwise, Corollary 3.3 implies

$$\begin{aligned} h^0(pD_{2n})-h^0(pD_{2n-1})\ge 2+h^0(pD_{2n-2})-h^0(pD_{2n-3})\ge 2n-1. \end{aligned}$$

and we are done.

(ii) The case $n=1$ is trivial. Assuming the assertion for $n-1$, we will prove

$$\begin{aligned} h^0(pD_{2n-1})-h^0(pD_{2n-2})\ge 1+h^0(pD_{2n-2})-h^0(pD_{2n-3}), \end{aligned}$$

(5)

and by (i) the right hand side is at least $2n-2$, which suffices for (ii). To prove the inequality (5), take $D=p\, Q_1$, $E=p\, D_{2n-3}$ and $F=p\, R$ with the point R satisfying $h^0(pR)=1+h^0((p-1)R)$ (see Corollary 3.2) and $Q_1$ a general point. We are going to verify the conditions $(1){-}(5)$ of Proposition 2.4 in the case of $m=0$. Conditions (1) and (2) are obvious by the property of R and generality of $Q_1$. For condition (3), the map induced by |pR| is separable for curves of genus $g>0$. We can choose $Q_1$ to be a point where the map is smooth.

For condition (4) we can choose $Q_1$ to be a non-base point of $|K_X-E|$ as it is non-empty. For condition (5), as $E=pD_{2n-3}$ with $n\ge 2$, we have for any point Q in $\mathrm{Supp}(E)$, $|pQ+pQ_1|$ is base point free due to Proposition 2.1. Then $|D+E+F|$ is base point free and by Proposition 2.4 we have

$$\begin{aligned} h^0(pD_{2n-1})-h^0(pD_{2n-2})\ge 1+h^0(pD_{2n-2})-h^0(pD_{2n-3})\ge 2n-2. \end{aligned}$$

(iii) For p odd, we let $n=(p+1)/2$ and apply (i) get $h^0(pD_{p+1})-h^0(pD_p)\ge p$. For $p=2$, we let $n=2$ and apply (ii) we also get $h^0(pD_{p+1})-h^0(pD_p)\ge p$. So we have $h^0(K_X-pD_{p})=h^0(K_X-pD_{p+1})$. In other words, for a general point Q, we see that $p\, Q$ lies in the base locus of $|K_X-pD_p|$. This can only happen when $h^0(K_X-pD_{p})=0$. Property (iv) follows by combining (i) and (ii). Property (v) follows by (iv) and Riemann–Roch. For property (vi), by (ii) and (iii), it is known that

$$\begin{aligned} h^0(pD_p)-h^0(pD_{p-1})&\ge p-1,~ h^0(K_X-pD_p)=0. \end{aligned}$$

We have

$$\begin{aligned} h^0(K_X-pD_{p-1})&=h^0(pD_{p-1})-1+g-p(p-1) \\&\le h^0(pD_p)-1+g-p(p-1)-(p-1)\\&=h^0(K_X-pD_p)+1=1. \end{aligned}$$

$\square$

Using the inequalities in Corollary 3.4, we can easily prove Theorem 1.1.

Proof (Proof of Theorem 1.1)

We estimate $g=h^0(K_X)$ by

$$\begin{aligned} h^0(K_X)&=\sum _{n=1}^{\left\lceil (p-1)/2\right\rceil }{(h^0(K_X-pD_{2n-2})-h^0(K_X-pD_{2n}))}+h^0(K_X-pD_{2\left\lceil (p-1)/2\right\rceil }) \\&\le \sum _{n=1}^{\left\lceil (p-1)/2\right\rceil }(2p-4n+3)+h^0(K_X-pD_{2\left\lceil (p-1)/2\right\rceil }) =p(p+1)/2. \end{aligned}$$

$\square$

Our approach to the case $\mathrm{rank}({\mathcal {C}})=2$ is similar, but there are differences due to the existence of special linear systems. We now give the analogue of Corollary 3.3.

Corollary 3.5

Let X be a non-hyperelliptic curve with $\mathrm{rank}({\mathcal {C}})=2$, and let $T_n$ be a general effective divisor of degree n and put $E=p\, T_n$. Let $Q_1$, $Q_2$, $Q_3$ be general points of X and put $D=E+\sum _{i=1}^3p\, Q_i$.

(1) Assume there exists R of X such that $h^0(p\, R)=1+h^0((p-1)\,R)$.

(a) If $h^0(K_X-E)\le 2$, then one has $h^0(D+pR)-h^0(D)=p$.

(b) If $h^0(K_X-E)\ge 3$, then one has

$$\begin{aligned} h^0(D+pR)-h^0(D)\ge h^0(E+pR)-h^0(E)+3. \end{aligned}$$

(2) If there does not exist such a point R, we choose points $R_1$,$R_2$ satisfying $h^0(\sum _{i=1}^2p\, R_i)=1+h^0(\sum _{i=1}^2p\, R_i-R_2)$ and let $\deg E \ge 2p$.

(a) If $h^0(K_X-E)\le 2$, then one has $h^0(D+\sum _{j=1}^2pR_j)-h^0(D)=2p$.

(b) If $h^0(K_X-E)\ge 3$, then one has

$$\begin{aligned} h^0\left( D+\sum _{i=1}^2pR_j\right) -h^0(D)\ge h^0\left( E+\sum _{j=1}^2pR_j\right) -h^0(E)+3. \end{aligned}$$

Note that in (2) we can choose such $R_1$ and $R_2$ by Lemma 3.1. The proof is similar to the proof of Corollary 3.3. But we point out that in the proof of part (2), instead of using the separable map induced by $|p\,R|$ in part (ii) of the proof of Corollary 3.3, we consider the map induced by $|p\,R_1+p\,R_2|$ with points $R_1$ and $R_2$. This map is separable, otherwise $\dim |R_1+R_2|\ge 1$, contradicting that X is non-hyperelliptic.

The following two corollaries are the analogues of Corollary 3.4.

Corollary 3.6

Let X be a non-hyperelliptic curve with $\mathrm{rank}({\mathcal {C}})=2$. Denote by $D_n$ a general divisor of degree n. If there exists a point R of X that $|p\, R|$ is base point free, then for any integer $n\ge 1$, one has

• (i) $~p\ge h^0(pD_{3n})-h^0(pD_{3n-1})\ge \min (3\,n-2,p)$.

• (ii)  For $1\le n \le \left\lceil (p+2)/3 \right\rceil$, one has

  $$\begin{aligned} 2\,p\ge h^0(pD_{3n-1})-h^0(pD_{3n-3})\ge \max (6\,n-7,0). \end{aligned}$$

• (iii)  $pD_{p+1}$ is non-special, i.e. $h^0(K_X-p\,D_{p+1})=0$.

• (iv)  For $1\le n \le \left[ (p+2)/3 \right]$, one has $h^0(pD_{3n})-h^0(pD_{3n-3})\ge 9\,n-9$.

• (v)  For $1\le n \le \left[ (p+2)/3 \right]$, one has

  $$\begin{aligned} h^0(K_X-pD_{3n-3})-h^0(K_X-pD_{3n})\le 3\,p-9\,n+9. \end{aligned}$$

• (vi)  $\,h^0(K_X-pD_{3\left\lceil (p-1)/3 \right\rceil })\le 3$.

Corollary 3.7

Let X be a non-hyperelliptic curve with $\mathrm{rank}({\mathcal {C}})=2$. Denote by $D_n$ a general divisor of degree n. If X does not possess a point R such that $|p\, R|$ is base point free, then for any integer $n\ge 1$, one has

• (i) $2\,p\ge h^0(pD_{3n})-h^0(pD_{3n-2})\ge \min (3\,n-2,2p)$.

• (ii) For $2\le n \le \left\lceil (2p+2)/3 \right\rceil$, one has

  $$\begin{aligned} 2\,p\ge h^0(pD_{3n-2})-h^0(pD_{3n-3})\ge 1. \end{aligned}$$

• (iii) $pD_{2p}$ is non-special, i.e. $h^0(K_X-pD_{2p})=0$.

• (iv) For $2\le n \le \left[ (2p+2)/3 \right]$, one has $h^0(pD_{3n})-h^0(pD_{3n-3})\ge 3\,n-1$. For $n=1$, one has $h^0(pD_{3})-h^0(pD_{0})\ge 1$.

• (v) For $2\le n \le \left[ (2p+2)/3 \right]$, one has

  $$\begin{aligned} h^0(K_X-pD_{3n-3})-h^0(K_X-pD_{3n})\le 3\,p-3\,n+1. \end{aligned}$$

  For $n=1$, one has $h^0(K_X)-h^0(K_X-pD_{3})\le 3\,p-1$.

• (vi) $\,h^0(K_X-pD_{3\left\lceil (2p-1)/3 \right\rceil })\le p-1$.

The proofs of two corollaries above are similar to the proof of Corollary 3.4 and therefore we omit these. The corollaries above now readily imply the proof of theorem in the case of $\mathrm{rank} ({\mathcal {C}})=2$.

Proof (Proof of Theorem 1.2)

• (1) If $|p\,R|$ is base point free, then by Corollary 3.6 we have

  $$\begin{aligned} h^0(K_X)&\le \sum _{n=1}^{\left\lceil (p-1)/3 \right\rceil }(h^0(K_X-pD_{3n-3})-h^0(K_X-pD_{3n}))+h^0(K_X-pD_{3\left\lceil (p-1)/3 \right\rceil })\\&\le \sum _{n=2}^{\left\lceil (p-1)/3 \right\rceil }(3\,p-9\,n+9)+1+3=p\,(p+3)/2. \end{aligned}$$

• (2) Otherwise, by Corollary 3.7 we have

  $$\begin{aligned} h^0(K_X)&\le \sum _{n=1}^{\left\lceil (2p-1)/3 \right\rceil }(h^0(K_X-pD_{3n-3})-h^0(K_X-pD_{3n}))+h^0(K_X-pD_{3\left\lceil (2p-1)/3 \right\rceil })\\&\le \sum _{n=2}^{\left\lceil (2p-1)/3 \right\rceil }(3\,p-3\,n+1)+3\,p-1+p-1=p\,(4\,p+1)/3. \end{aligned}$$

$\square$