4
$\begingroup$

A search of the literature reveals that for a curve $C$ of genus $\geq 2$, determining the effective cone of $C \times C$ is hard. My question is this: do we know a single example of a curve $C$ of genus $\geq 2$ for which we understand the effective cone of $C \times C$ and its image in $\text{Num}(C \times C)$? In particular, when can we say that the image in $\text{Num}(C \times C)$ is finitely generated?

Edit: Since there haven't been any answers to my question about finite generation, perhaps I can be more specific. Let $k = \mathbb{C}$. If $C = \mathbb{V}(x^n + y^n + z^n) \subset \mathbb{P}^2_k$, are there any techniques to calculate the effective cone of $C \times C$ when $n \geq 4$?

I ask this because if $k$ is a field of characteristic $p > 0$, it's well-known that this cone is not finitely generated since we have infinitely many extremal curves $\Gamma_q := \{(u,v) | \text{ Frob}_q(u) = v \}$ where $q = p^r$.

$\endgroup$

2 Answers 2

4
$\begingroup$

Since I asked this question several months ago, I wrote a paper proving that the Mori cone of $C \times C$ for a smooth projective complex curve $C$ of genus $g \geq 2$ is not polyhedral. I thought I'd include a link to that for completeness.

http://arxiv.org/abs/1502.06061

$\endgroup$
1
$\begingroup$

You should like at Section $5$ of this paper

http://www.ams.org/journals/tran/1993-337-01/S0002-9947-1993-1149124-5/S0002-9947-1993-1149124-5.pdf

for bounds on the effective an ample cones of $C\times C$.

$\endgroup$
1
  • $\begingroup$ Question edited. Also, Kouvidakis' paper is about $\text{Sym}^2(C)$, not $C \times C$. Is there an easy result that one can deduce from $\text{Sym}^2(C)$? Since $N^1(C \times C)$ is generically three-dimensional, it might be harder. $\endgroup$ Aug 10, 2014 at 1:12

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.