Some Results on Seshadri constants on Surfaces of general type

TL;DR

This paper investigates Seshadri constants on surfaces of general type, providing classifications for certain multi-point constants and proving rationality of global constants on product surfaces.

Contribution

It offers new classifications of multi-point Seshadri constants and establishes the rationality of global Seshadri constants on product surfaces of curves.

Findings

Classified possible values of multi-point Seshadri constants between 0 and 1/r.

Proved global Seshadri constants are rational on surfaces of the form C×C.

Extended understanding of Seshadri constants on surfaces of general type.

Abstract

We prove two new results for Seshadri constants on surfaces of general type. Let $X$ be a surface of general type. In the first part, inspired by \cite{B-S}, we list the possible values for the multi-point Seshadri constant $\varepsilon(K_X,x_1,x_2,...,x_r)$ when it lies between $0$ and $1/r$, where $K_X$ is the canonical line bundle on $X$. In the second part, we assume $X$ of the form $C \times C$, where $C$ is a general smooth curve of genus $g \geq 2$. Given such $X$ and an ample line bundle $L$ on $X$ with some conditions on it, we show that the global Seshadri constant of $L$ is a rational number.

Equations

$$\varepsilon(X,L,x):=\inf\limits_{\begin{subarray}{c}x\in C\end{subarray}}\frac{L\cdot C}{{\rm mult}_{x}C},\$$

$$\epsilon (X,L,1):=\underset{x\in X}{\sup }\epsilon (X,L,x)\text{and}\epsilon (X,L):=\underset{x\in X}{\inf }\epsilon (X,L,x).$$

$$0<\epsilon (X,L)\le \epsilon (X,L,x)\le \epsilon (X,L,1)\le \sqrt{L^2}.$$

$$\epsilon (X,L,x_1,x_2,...,x_r):=\underset{C\cap \{x_1,x_2,...,x_r\}\ne \varphi }{\inf }\frac{L\cdot C}{\sum _{i=1}^{i=r}mul{t}_{x_i}C},$$

$$\epsilon (X,L,x_1,x_2,...,x_r)\le \sqrt{\frac{L^2}{r}}.$$

$$\epsilon (X,L,r):=\underset{\begin{array}{c}{x}_1,x_2,...,x_r\in X\end{array}}{\max }\epsilon (X,L,x_1,x_2,...,x_r).$$

$${\varphi }_{mL}:X\text{dasharrow}\mathbb{P}(H^0(X,mL)).$$

$$\epsilon (X,K_X,x_1,x_2,...,x_r)=\{\begin{array}{ll}\frac{1}{r+1}\text{or}\frac{2}{5}& \text{if}r=2,\\ \frac{1}{r+1}\text{or}\frac{1}{r+2}& \text{if}3\le r<10,\\ \frac{1}{r+1}\text{or}\frac{1}{r+2}\text{or}\frac{1}{r+3}& \text{if}r\ge 10.\end{array}$$

$$\epsilon (X,K_X,x_1,x_2,...,x_r)=\frac{K_X\cdot C}{m},$$

$$0\le P_a(\tilde{C})$$

$$\Rightarrow \sum _{i=1}^{i=r}\frac{m_i(m_i-1)}{2}$$

$$\Rightarrow \left(\frac{1}{r}{m}^2-m\right)$$

$$\Rightarrow m^2-rm$$

$${\varphi }_{r,d}(dr+j):=r^2{d}^2+j^2+2drj-r^{2}d-r{d}^2-(d+2)r-rj.$$

$${\varphi }_{r,d}^{\prime }(rd+j)=r^2(2d-1)-r(2d+1-2j)$$

$$(F_1{)}^2$$

$$F_1\cdot F_2$$

$${\delta }^2$$

$$L\cdot F_1$$

$$L\cdot F_2$$

$$L\cdot \delta$$

$$L^2$$

$$\text{either}{a}_1^2\le 2a_1{a}_2$$

$$\iff \text{either}{a}_1\le 2a_2$$

$$2a_1>a_1>2a_2>a_2,$$

$$L\cdot F_2\le \sqrt{L^2}$$

$$\iff$$

$$L\cdot \delta =a_1+a_2-a_3(2g-2)>0$$

$$\Rightarrow$$

$$L\cdot F_2=a_1+a_3\le \sqrt{L^2}.$$

$$L\cdot F_1=a_2+a_3>0\text{and}L\cdot F_2=a_1+a_3>0.$$

$$a_3>-\frac{k}{k+1}{a}_1,$$

$$k:=\lceil \frac{|a_3|/a_1}{1-|a_3|/a_1}\rceil .$$

$$a_1\left(\frac{2g{k}^2+2k+1}{2(k+1)}\right)$$

$$\iff a_1^2\left(\frac{2g{k}^2+2k+1}{(k+1{)}^2}\right)$$

$$\iff a_1^2+{\left(\frac{k}{k+1}\right)}^2{a}_1^2(2g-1)$$

$$a_1^2+a_3^2(2g-1)$$

Full text

Author was partially supported by a grant from Infosys Foundation

1. introduction

Seshadri constants have turned out to be a powerful tool to measure local positivity of an ample line bundle on a projective variety. They were defined by Demailly using the Seshadri criterion of ampleness for a line bundle [4]. Since then, the area has emerged to be quite active with computing and bounding the Seshadri constants becoming an active area of research. For a detailed survey and the typical nature of work, we refer to [1, 5, 6, 7, 8, 9].

Let $X$ be a smooth complex projective surface and let $L$ be a line bundle on $X$. Given a point $x\in X$, Seshadri criterion for ampleness [10] says that $L$ is an ample line bundle on $X$ if and only if there exists a positive real number $\varepsilon$ such that $L\cdot C\geq\varepsilon\cdot mult_{x}C$ for all $x\in C$. Here, “$mult_{x}C$" denotes the multiplicity of $C$ at the point $x$. Given $X$ and $L$ as above, the Seshadri constant $\varepsilon(X,L,x)$ of $L$ at a point $x\in X$ [11] is defined as

[TABLE]

where the infimum is taken over all irreducible and reduced curves $C$ passing through $x$. Now, it is easy to see that $L$ is ample if and only if $\varepsilon(X,L,x)>0$ for all $x\in X$. There are various directions in which one can study Seshadri constants. For a comprehensive survey, we refer to [1].

Given a smooth complex projective surface $X$ and an ample line bundle $L$ on $X$, it is not difficult to see that $0<\varepsilon(X,L,x)\leq\sqrt{L^{2}}$ for every $x\in X.$ Thus, it makes sense to define:

[TABLE]

These satisfy the following inequalities:

[TABLE]

It is known that $\varepsilon(X,L,1)$ is attained at a very general point $x\in X$ [12]. Further, if $\varepsilon(X,L,1)<\sqrt{L^{2}}$, then there exists a reduced and irreducible curve $C\subset X$ containing a very general point $x$ such that $\varepsilon(X,L,1)=\varepsilon(X,L,x)=\frac{L\cdot C}{mult_{x}C}$. Therefore, the Seshadri constant is a rational number in this situation. Consequently, for an irrational Seshadri constant, $\varepsilon(X,L,x)$ must be equal to $\sqrt{L^{2}}$ and, $L^{2}$ must be non-square. However, there is no known example of a triple $(X,L,x)$ which gives an irrational Seshadri constant.

On the other hand, $\varepsilon(X,L)$ is computed at some special points $x\in X$. Therefore, in order to compute $\varepsilon(X,L)$ one needs to find some information about the curves passing through $x\in X$. In other words, geometry of $X$ near that point is important.

1.1. Multi-point Seshadri constants

Let $X$ be a smooth complex projective surface and $L$ be an ample line bundle on $X$. Let $r\geq 1$ be an integer and $x_{1},x_{2},...,x_{r}\in X$ be $r$ distinct points. Then, the multi-point Seshadri constant of $L$ at $x_{1},x_{2},...,x_{r}\in X$ is defined as:

[TABLE]

where the infimum is taken over all reduced and irreducible curves $C\subset X$ passing through at least one of the points $x_{1},x_{2},...,x_{r}\in X$. A well known upper bound for the multi-point Seshadri constant is

[TABLE]

The Seshadri constant is said to be sub-maximal if the above inequality is strict, and in that case it is computed by a curve $C\subset X$ (i.e., $\varepsilon(X,L,x_{1},x_{2},...,x_{r})=L\cdot C/\sum\limits_{i=1}^{i=r}{mult_{x_{i}}C}$), which is known as the Seshadri curve. See [2, Proposition 1.1] for a proof of their existence in the single point case which generalizes easily to the multi-point case.

One can then define:

[TABLE]

It is known that $\varepsilon(X,L,r)$ is attained at very general points $x_{1},x_{2},...,x_{r}\in X$, i.e., there exists a set $U\subset X^{r}:=X\times X\times...\times X$ which is the complement of a countable union of proper closed subsets of $X^{r}$, such that $\varepsilon(X,L,r)=\varepsilon(X,L,x_{1},x_{2},...,x_{r})$ for all $(x_{1},x_{2},...,x_{r})\in U$. It is conjectured that, $\varepsilon(X,L,r)$ is equal to $\sqrt{{L^{2}}/{r}}$ for large $r$ [14]. In fact, the Nagata-Biran-Szemberg Conjecture predicts exactly when it happens. It says that the multi-point Seshadri constant at a very general set of points is maximal when $r\geq k_{0}^{2}L^{2}$, where $k_{0}$ is the smallest integer such that the linear system $|k_{0}L|$ contains a smooth non-rational curve.

In this article, we study some of the questions discussed above on surfaces of general type. Note that surfaces of general type [3] are minimal surfaces of Kodiara dimension two (see def. (2.1)). Not much is known about these surfaces compared to surfaces of Kodaira dimension $1$, [math] or $-\infty$. Here, we have considered a class of such surfaces of the form $C\times C$, where $C$ is a smooth curve of genus at least two, and have answered some of the questions about Seshadri constants.

This paper is divided into two parts. In §(2), we prove a result about the multi-point Seshadri constant of canonical line bundle on a surface of general type. In §(3), we consider surfaces of general type of the form $C\times C$, where $C$ is a general member of the moduli of smooth curves of genus $g\geq 2$ and answer some of the questions discussed above.

2. Multi-point Seshadri constants on surfaces of general type

Let $X$ be a smooth complex projective variety, and let $L$ be a line bundle on $X$. Consider the linear system $|mL|$ for $m\in\mathbb{N}$. The global sections of $mL$ defines a rational map

[TABLE]

Clearly $dim(\phi_{mL}(X))\leq dim(X)$.

**Notation: **

$\kappa(X,L):=\max\{dim(\phi_{mL}(X)):m\in\mathbb{N}\}$.

Definition 2.1**.**

Given a smooth complex projective variety $X$ with canonical divisor $K_{X}$, the Kodaira dimension of $X$ is defined as $\kappa(X,K_{X})$.

Definition 2.2**.**

A smooth complex algebraic surface $X$ is said to be of general type if the Kodaira dimension $\kappa(X)=2$.

One defines a line bundle $L$ on a smooth complex projective variety $X$ to be big if $\kappa(X,L)=dim(X)$. Therefore, a surface of general type is a surface whose canonical divisor is big. The following theorem is a characterisation for a nef line bundle to be big [11].

Theorem 2.3**.**

Let $X$ be an irreducible projective variety of dimension n and $L$ be a nef line bundle on $X$. Then $L$ is big if and only if its top self-intersection is strictly positive, i.e., $(L^{n})>0$.

Motivated by [2, Theorem 1], we prove the following:

Theorem 2.4**.**

Let $X$ be a surface of general type and $K_{X}$ be the canonical line bundle on $X$. If $K_{X}$ is big and nef and $x_{1},x_{2},...,x_{r}\in X$ are $r\geq 2$ points, then we have the following.

1. (1)

$\varepsilon(X,K_{X},x_{1},x_{2},...,x_{r})=0\Leftrightarrow$* at least one of $x_{i}$ lies on one of the finitely many (-2)-curves on $X$.* 2. (2)

If $0<\varepsilon(X,K_{X},x_{1},x_{2},...,x_{r})<\frac{1}{r}$, then

[TABLE]

Proof.

The proof of $(1)$ uses the same technique as the case of the single point Seshadri constant in [2]. Since $K_{X}$ is big and nef, its self-intersection is strictly positive, i.e., $K_{X}^{2}>0$.

$\Rightarrow$: Let $C\subset X$ be a smooth curve passing through at least one of the points $x_{1},x_{2},...,x_{r}\in X$ with multiplicities $m_{1},m_{2},...,m_{r}$, such that $0=\varepsilon(X,K_{X},x_{1},x_{2},...,x_{r})=K_{X}\cdot C/\sum\limits_{i=1}^{i=r}{m_{i}}$. This gives $K_{X}\cdot C=0$. Using the Hodge Index Theorem and the fact that $K_{X}^{2}>0$, we get $C^{2}<0$. Since $K_{X}$ is nef, there are no $(-1)$-curves in $X$, therefore $C^{2}=-2$. Using adjunction formula we conclude that the genus of $C$ is [math], and hence $C$ is a rational curve.

$\Leftarrow$: Conversely, suppose some $x_{i}$ lies on a $(-2)$-curve $C$, then using the adjunction formula and the fact that the arithmetic genus of $C$ is [math], we find that $K_{X}\cdot C=0$. Hence $\varepsilon(X,K_{X},x_{1},x_{2},...,x_{r})=\frac{K_{X}\cdot C}{\sum\limits_{i=1}^{i=r}{mult_{x_{i}}C}}=0$.

$(2)$ Let $\varepsilon(X,K_{X},x_{1},x_{2},...,x_{r})<\frac{1}{r}$ which in turn is less than $\sqrt{\frac{K_{X}^{2}}{r}}$, so that by a generalized statement of [2, Proposition 1.1] for the multi-point case, there exists a reduced and irreducible curve $C$ computing $\varepsilon(X,K_{X},x_{1},x_{2},...,x_{r}).$ Let $C$ be a reduced and irreducible curve in $X$ passing through at least one of the points $x_{1},x_{2},...,x_{r}$ with multiplicities $m_{1},m_{2},...,m_{r}$ such that

[TABLE]

where $m:=\sum\limits_{i=1}^{i=r}{m_{i}}$. Put $K_{X}\cdot C=d$. Notice that $\frac{d}{m}<\frac{1}{r}$ which gives $m>dr$. Now, using the positivity of $K_{X}^{2}$ and the Hodge Index Theorem, we get $C^{2}\leq K_{X}^{2}C^{2}\leq(K_{X}\cdot C)^{2}=d^{2}$. Let $\tilde{C}$ be the normalization of $C$. Then,

[TABLE]

We see that equation (2.1) implies the inequality (2.2). Therefore, it is enough to find out when the inequality (2.2) holds. We show that the possible choices of $d$ and $m$ satisfying the above conditions are as stated in the statement of the theorem.

Put $\phi_{r,d}(m):=m^{2}-rm-r(2+d^{2}+d)$.

**Claim: **

$\phi_{r,d}(m)<0$ $\Longrightarrow$ $d=1,2$ and $m=r+1,r+2,r+3\;\text{and}\;5$ with some conditions on $r$.

Since $m>dr$, substituting $m=dr+j$ in $\phi_{r,d}(m)$, we get

[TABLE]

$d=1$**: **

\phi_{r,1}(r+j)=j^{2}+rj-4r\;\begin{cases}<0&\text{if j=1$and$r\geq 2}\cr<0&\text{if j=2$and$r\geq 3}\cr<0&\text{if j=3$and$r\geq 10}\cr>0&\text{otherwise}\cr\end{cases}

$d=2$**: **

\phi_{r,2}(2r+j)=2r^{2}+j^{2}+3rj-8r\;\begin{cases}<0&\text{if j=1$and$r=2}\cr>0&\text{otherwise}\cr\end{cases}

$d\geq 3$**: **

$\phi_{r,d}(rd+j)>0$ for $r\geq 2$.

In order to see the last statement, it is sufficient to show that $\phi_{r,3}(3r+j)>0$ for $r\geq 2$ and the derivative of $\phi_{r,d}$ with respect to $d$ is positive. This will imply that $\phi_{r,d}$ is an increasing function of $d$ and hence positive for all $d\geq 3,r\geq 2$. The first condition is easily checked. The second condition is also satisfied since differentiating $\phi_{r,d}$ with respect to $d$ gives

[TABLE]

which is always positive whenever $r\geq 2$. ∎

3. Surface of general type of the form $C\times C$

Let $C$ be a smooth complex projective curve of genus $g\geq 2$ and consider a surface $X=C\times C$. Let $F_{1}$ and $F_{2}$ be fibres corresponding to the two projections from $C\times C\longrightarrow C$ and let $\delta$ be the diagonal. Assume that $C$ is a general member of the moduli of smooth curves of genus $g$, where $g\geq 2$. Then, it is known that the Néron-Severi group $NS(X)$ is spanned by $F_{1},F_{2}$ and $\delta$ [11, 1.5B]. Intersections among them is governed by the following formulae:

[TABLE]

Let $K_{X}$ be the canonical divisor of $X$. Then, it can be checked that $K_{X}^{2}=8(g-1)^{2}$ is always positive [10].

We consider $X$ defined as above and compute the Seshadri constant of an ample line bundle $L$ on $X$. Let $L\equiv_{num}a_{1}F_{1}+a_{2}F_{2}+a_{3}\delta$, where $a_{1},a_{2},a_{3}\in\mathbb{Z}$ and “$\equiv_{num}$” represents the $numerical$ $equivalence$. Since $L$ is ample, we have

[TABLE]

3.1. Results about $\varepsilon(L)$

In this section we partially answer the question about the rationality of $\varepsilon(L)$ [13, Question 1.6]. In other words, under some conditions on $a_{1},a_{2}$ and $a_{3}$ we address the question of rationality in affirmative. Following is our main theorem.

Theorem 3.1**.**

Let $X=C\times C$, where $C$ is a general member of moduli of smooth curves of genus $g\geq 2$. Let $L\equiv_{num}a_{1}F_{1}+a_{2}F_{2}+a_{3}\delta$ be an ample line bundle satisfying any of the following conditions on $a_{1},a_{2}$ and $a_{3}$.

1. (1)

$a_{3}=0$, 2. (2)

$a_{3}>0$, $a_{1}\leq a_{2}$ and $a_{1}^{2}+a_{3}^{2}<2a_{1}a_{2}$, 3. (3)

$a_{3}>0$, $a_{2}\leq a_{1}$ and $a_{2}^{2}+a_{3}^{2}<2a_{1}a_{2}$, 4. (4)

$a_{3}<0$* and $a_{2}\geq\left(\frac{2gk^{2}+2k+1}{2(k+1)}\right)\cdot a_{1}$, where $k=\lceil\frac{|a_{3}|/a_{1}}{1-|a_{3}|/a_{1}}\rceil$ or* 5. (5)

$a_{3}<0\;\text{and}\;a_{1}\geq\left(\frac{2gl^{2}+2l+1}{2(l+1)}\right)\cdot a_{2}$, where $l=\lceil\frac{|a_{3}|/a_{2}}{1-|a_{3}|/a_{2}}\rceil$.

Then $\varepsilon(L)\in\mathbb{Q}$.

Proof.

$(1)$ Assume $a_{3}=0$, then we have $L\equiv_{num}a_{1}F_{1}+a_{2}F_{2}$. In this case, we show that either $L\cdot F_{2}\leq\sqrt{L^{2}}$ or $L\cdot F_{1}\leq\sqrt{L^{2}}$. This is equivalent to show that

[TABLE]

Notice that, when $a_{1}>2a_{2}$, we get

[TABLE]

implying that the statement (3.6) always holds.

$(2)$ Let $a_{3}>0$, $a_{1}\leq a_{2}$ and $a_{1}^{2}+a_{3}^{2}<2a_{1}a_{2}$. Then, we show that $L\cdot F_{2}\leq\sqrt{L^{2}}$. Notice that

[TABLE]

Now, since $L$ is ample, we have

[TABLE]

It is easy to see that the equation (3.7) follows from the hypothesis and the equation (3.9).

$(3)$ The proof follows similar to that of (2).

$(4)$ Let $a_{3}<0$ and $a_{2}\geq\left(\frac{2gk^{2}+2k+1}{2(k+1)}\right)a_{1}$, where $k=\lceil\frac{|a_{3}|/a_{1}}{1-|a_{3}|/a_{1}}\rceil$. We will show that

[TABLE]

It suffices to show that the equation (3.8) holds. Since $L$ is ample, we get

[TABLE]

This implies that, $a_{3}$ can at the very least be $-a_{1}$, i.e., $a_{3}>-a_{1}>-a_{2}$. Thus, there must exist a positive integer $k$ such that

[TABLE]

since $-(k/k+1)a_{1}$ converges to $-a_{1}$. Choose the least such positive $k$ for which the above inequality holds. That is,

[TABLE]

Here, $\lceil x\rceil$ represents the least integer greater than or equal to $x$. We have the following

[TABLE]

Therefore, the following holds:

[TABLE]

Where the first and last inequalities hold by (3.12) and the fact that $a_{3}<0$, while the second inequality follows from (3.15). Therefore, inequality (3.8) holds.

(5) The proof is similar to that of (4). ∎

Example 3.2**.**

We give an example to show the occurrence of case (4). Let $X=C\times C$ be a surface of general type, where $C$ is a smooth curve of genus $g$. Let $L\equiv_{num}a_{1}F_{1}+a_{2}F_{2}+a_{3}\delta$ be an ample line bundle on $X$. Assume that $a_{2}>a_{1}$ as in the hypothesis of (4). Therefore, $a_{2}>a_{1}>|a_{3}|>0$.

Note that $k=\lceil\frac{|a_{3}|}{a_{1}-|a_{3}|}\rceil,$ and, in general, we have $1\leq k\leq|a_{3}|$. When $a_{1}\geq 2|a_{3}|$, we have $k=1$. Then, the condition on $a_{2}$ becomes

[TABLE]

So for an ample line bundle $L\equiv_{num}a_{1}F_{1}+a_{2}F_{2}+a_{3}\delta$ with $a_{3}<0$, $a_{1}\geq 2|a_{3}|$, and $a_{2}>\left(\frac{2g+3}{4}\right)a_{1}$, we have $\varepsilon(L)\in\mathbb{Q}$.

For example, fix $g=2$ and take $a_{3}=-10$. Then if $a_{1}=20$, we get the least value of $k$ i.e., 1. In this case, we require $a_{2}\geq(7/4)a_{1}=35$. But when $a_{1}=11$, we get the highest value of $k$ i.e., 10. So we require $a_{2}\geq(421/22)a_{1}=382.72$.

Now we prove the following theorem for $\varepsilon(K_{X},r)$, where $X=C\times C$ as in the above theorem. The primary motivation behind this theorem is [7].

Theorem 3.3**.**

Let $X=C\times C$, where $C$ is a general member of the moduli of smooth curves of genus $g\geq 2$. Let $K_{X}$ be the canonical line bundle on $X$ and $r\geq K_{X}^{2}$ be an integer. Then either

[TABLE]

or $\varepsilon(X,K_{X},r)$ is computed by a curve $C_{1}$ numerically equivalent to $a(F_{1}+F_{2})$ (for some $a\in\mathbb{N}$) passing through $r$ very general points with multiplicity one at each point. In other words,

[TABLE]

Proof.

Suppose

[TABLE]

Then, there exists an effective curve $C_{1}\subset X$ passing through $s\leq r$ very general points with multiplicities one each [7], such that

[TABLE]

By [7, Remark 2.4], we get $C_{1}^{2}<s.$ Also, since $C_{1}$ is a curve in $X$ passing through $s\leq r$ very general points with multiplicities $m_{1}\geq m_{2}\geq...\geq m_{s}>0$, then by Xu’s lemma [15],

[TABLE]

Thus, we have $C_{1}^{2}=s-1$. We will show that $C_{1}$ is numerically equivalent to $a(F_{1}+F_{2})$ for some $a\in\mathbb{N}$.

• $Case$ 1

: $s=1$

In this case, we have $\varepsilon(X,K_{X},r)=K_{X}\cdot C_{1}\geq 1>\sqrt{\frac{r+2}{r+3}}\sqrt{\frac{K_{X}^{2}}{r}}$ since $r\geq K_{X}^{2}$. This is a contradiction to our assumption.

• $Case$ 2

: $2\leq s\leq r-1$

Notice that

[TABLE]

Using Hodge Index Theorem, we obtain $(K_{X}\cdot C_{1})^{2}\geq(s-1)K_{X}^{2}$ and hence equation (3.17) follows if we prove

[TABLE]

This is true for $r\geq 4$. To see this, it is enough to check the inequality at the maximal possible value of $s$, i.e., at $s=r-1$:

[TABLE]

This holds for $r\geq 4$. By hypothesis $r\geq K_{X}^{2}=8(g-1)^{2}\geq 8$. So we again arrive at a contradiction to our assumption.

• $Case$ 3

: $s=r$

Notice that, the equation (3.17) follows if we prove $(K_{X}\cdot C_{1})^{2}\geq(C_{1}^{2}+\frac{1}{3})K_{X}^{2}=\left(r-\frac{2}{3}\right)K_{X}^{2}$, because we have the following

[TABLE]

However, the last inequality holds for $r\geq 6$. Now to see $(K_{X}\cdot C_{1})^{2}\geq(C_{1}^{2}+\frac{1}{3})K_{X}^{2}$, we start by putting $C_{1}\equiv_{num}a_{1}F_{1}+a_{2}F_{2}+a_{3}\delta$ for some $a_{1},a_{2},a_{3}\in\mathbb{Z}$ and $L:=F_{1}+F_{2}$. We know that $K_{X}=2(g-1)L$ [10], so it is enough to show that

[TABLE]

This clearly holds when $a_{3}\neq 0$. In the case $a_{3}=0$, we see that the equation $a_{1}^{2}+a_{2}^{2}\geq 2a_{1}a_{2}+2/3$ does not hold only when $a:=a_{1}=a_{2}$. In the latter case, $C_{1}\equiv_{num}a(F_{1}+F_{2})$ is a curve passing through $r$ points with multiplicity one each such that

[TABLE]

∎

Now, for a line bundle of the form $L\equiv_{num}aF_{1}+bF_{2}$ with $a,b>0$ we explicitly compute the Seshadri constants of $L$ at one or two points.

Theorem 3.4**.**

Let $X=C\times C$, where $C$ is a smooth curve of genus $g\geq 2$ and let $L\equiv_{num}aF_{1}+bF_{2}$ be an ample line bundle on $X$. Then $\varepsilon(L,x)=\min\{a,b\}$ for every $x\in X$.

Proof.

Since a fibre numerically equivalent to $F_{1}$ and $F_{2}$ passes through every point $x\in X$, we get

[TABLE]

Now, let $C_{1}$ be any curve in $X$ (not numerically equivalent to $F_{1}$ and $F_{2}$) passing through $x$ with multiplicity $m$. Then, by Bézout’s theorem we obtain

[TABLE]

for $i=1$ and $2$. Therefore, notice that

[TABLE]

Hence, we get $\varepsilon(L,x)=\min\{a,b\}.$ ∎

Theorem 3.5**.**

Let $X=C\times C$, where $C$ is a smooth curve of genus $g\geq 2$ and let $L\equiv_{num}aF_{1}+bF_{2}$ be an ample line bundle on $X$. Then

[TABLE]

Proof.

Let $C_{1}$ be a curve not numerically equivalent to $F_{1}$ and $F_{2}$ and passing through $x_{1}$ and $x_{2}$ with multiplicity $m_{1}$ and $m_{2}$ respectively. Since there is a fibre numerically equivalent to $F_{1}$ and $F_{2}$ passing through every point of $X$, by Bézout’s theorem we get

[TABLE]

This gives $C_{1}\cdot F_{1}\geq m$ and $C_{1}\cdot F_{2}\geq m$ where $m:=\max\{m_{1},m_{2}\}$. Now

[TABLE]

since $2m\geq m_{1}+m_{2}$. Now, if both the points $x_{1}$ and $x_{2}$ lie either on a fibre $F_{1}$ or on a fibre $F_{2}$, then we have

[TABLE]

However, $\min\{a,b\}\geq\min\left\{a,\frac{b}{2}\right\}$ and $\min\{a,b\}\geq\min\left\{\frac{a}{2},b\right\}$. Therefore, we get

[TABLE]

In case both the points $x_{1}$ and $x_{2}$ do not lie on the same fixed fibre, then we get

[TABLE]

Hence, we obtain $\varepsilon(L,x_{1},x_{2})=\min\{a,b\}.$ ∎

Remark 3.6**.**

When $X$ is as in the above two theorems, i.e., of the form $C\times C$, the canonical divisor $K_{X}$ of $X$ is given by $p_{1}^{*}(K_{C})\otimes p_{2}^{*}(K_{C})$ where $p_{1}$ and $p_{2}$ are the two natural projections from $C\times C\longrightarrow C$. Since $deg(K_{C})$ is $2(g-1)$, $K_{X}$ is numerically equivalent to $2(g-1)(F_{1}+F_{2})$. Hence the above two theorems apply to $K_{X}$.

$\mathbb{Acknowledgements}:$ I would like to thank my advisor Prof. Krishna Hanumanthu for his constant encouragement and many useful remarks which were helpful in writing this paper. I would also like to thank Prof. Tomasz Szemberg for giving me the idea of studying surfaces of general type of the form $C\times C$ and his generous support when I visited the Pedagogical University of Cracow. Lastly, I want to thank Prof. D. S. Nagaraj for giving many suggestions which improved this paper.

References