1
vote
0answers
9 views

is it possible to express the moduli of ppav's using torelli loci?

This is a probably vague question from an outsider: It is well known that it is possible to embed "the" moduli space of curves $M_g$ with fixed genus g into the moduli space of principally polarized ...
3
votes
1answer
125 views

Picard group and jacobian of a singular curve

I found somewhere written that the jacobian of a reducible curve is the product of the jacobians, which seems quite reasonable to me. Where can I find some proof of it and a description of the Picard ...
4
votes
1answer
102 views

Prym variety associated to an étale cover of degree 2 of an hyperelliptic curve.

In view of this question, I have an additional question. The situation is as follows. Let $C$ be the hyperelliptic curve over $\mathbb{C}$, which is given on an affine by the equation $y^2 = x^5 +1 ...
4
votes
1answer
54 views

What is this cycle on the Jacobian of a curve?

Let $C$ be a curve and $J$ it's Jacobian. There is the standard Abel-Jacobi map $a:C\rightarrow J$ which is given by $Q\rightarrow Q-P$ for some fixed point $P$ (here I am regarding $J$ as the degree ...
2
votes
0answers
45 views

Specifying the real multiplication on a curve

Suppose we have a curve $C$ of genus two over an algebraically closed field of characteristic zero with a given Weierstrass point $p_0$, furthermore we assume this curve to have real multiplication by ...
10
votes
1answer
159 views

Why does Mumford want to avoid “reduction to Jacobians”?

In the introduction to his Abelian Varieties book, David Mumford writes: I don't believe the word "Jacobian" is ever used in this book. Rather stubbornly I wanted to prove that the theory of ...
3
votes
0answers
50 views

Is there a construction known for associating a K3 surface to a curve or cover of curves

Let $X$ be a curve of genus at least two. Then one can associate an abelian variety to $X$; this is the Jacobian. Let $X\to Y$ be a double cover of curves. Then we can associate an abelian variety to ...
2
votes
0answers
31 views

Are there generalizations of Prym varieties to higher dimensions

Prym varieties are abelian varieties that are associated to a double cover of algebraic curves. Can we also associate an abelian variety to a double cover of algebraic surfaces in a reasonable way? ...
4
votes
0answers
133 views

If $X\times C$ is isomorphic to $Y\times C$, does it follow that $X$ is isomorphic to $Y$

Let $k$ be a field. Let $X$, $Y$ and $C$ be varieties over $k$ such that $X\times_k C $ is isomorphic to $Y\times_k C$. Assume that $C$ is a curve. (The varieties $X$, $Y$ and $C$ are assumed to be ...
6
votes
1answer
111 views

Is $M_g$ NEVER proper? And why does $T_g$ contain products?

Work over an algebraically closed field ($\mathbb C$, if you prefer) and fix $g\geq 2$. By $M_g$ I mean, of course, the moduli space of smooth projective curves of genus $g$. I know it is in general ...
3
votes
1answer
79 views

(Non)-Isomorphic Jacobians

I've started reading Hindry-Silverman's Diophantine Geometry and I got a little ahead of myself. Non-isomorphic elliptic curves have non-isomorphic Jacobians, that is because elliptic curves are ...
4
votes
1answer
189 views

Jacobian of a curve

Let $C$ be a curve and $J$ be its Jacobian. What is the relation between $H^1(C,\mathcal{O}_C)$ and $H^1(J,\mathcal{O}_J)$ ? Can someone point me to an easy reference for this subject?