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6
votes
2answers
306 views

Calculate reduction of Jacobian of hyperelliptic curve

Suppose I have a hyperelliptic curve of genus $2$ over $\mathbb Q$. I want to get information about its Jacobian reduction at prime $p$ (especially, in case $p=2$). Also I'm interesting in the group ...
18
votes
7answers
3k views

Links between Riemann surfaces and algebraic geometry

I'm taking introductory courses in both Riemann surfaces and algebraic geometry this term. I was surprised to hear that any compact Riemann surface is a projective variety. Apparently deeper links ...
21
votes
2answers
1k views

The work of E. Artin and F. K. Schmidt on (what are now called) the Weil conjectures.

I was reading Dieudonne's "On the history of the Weil conjectures" and found two things that surprised me. Dieudonne makes some assertions about the work of Artin and Schmidt which are no doubt ...
10
votes
3answers
2k views

Vector bundles on $\mathbb{P}^1\times\mathbb{P}^1$

I have a question about vector bundles on the algebraic surface $\mathbb{P}^1\times\mathbb{P}^1$. My motivation is the splitting theorem of Grothendieck, which says that every algebraic vector bundle ...
9
votes
1answer
257 views

Showing that $2c_1(f_*\mathscr O_X)=-f_*R_f$ on curves, maybe by local fields

I originally asked this question on Mathematic StackExchange, but it did not seem to be attracting any attention, so now I am trying mathoverflow. I hope it is not too simple or unappropriate a ...
5
votes
2answers
838 views

What does the Jacobian of a curve tell us about the curve?

A natural object in the study of curves is the Jacobian of a curve. What are some natural geometric properties of the curve that the Jacobian encapsulates? In other words, what can the Jacobian tell ...
8
votes
2answers
2k views

Kodaira-Spencer map in a concrete instance

Let $\pi:X_{\epsilon} \rightarrow \Delta$ be a family of (say smooth) projective plane curves parametrized by $\Delta:=\operatorname{Spec}(k[\epsilon])$, and let $X=X_0$ be the closed fiber. Suppose ...
6
votes
1answer
342 views

Understanding of Tamagawa numbers of hyperelliptic curve

One's can find following definition of tamagawa numbers in Dino Lorenzini paper "Torsion and Tamagawa numbers": Let $K$ be any discrete valuation field with ring of integers $O_K$ , uniformizer ...
8
votes
1answer
371 views

what is the maximum number of rational points of a curve of genus 2 over the rationals

Conjecturally, there exists an integer $n$ such that the number of rational points of a genus $2$ curve over $\mathbf{Q}$ is at most $n$. (This follows from the Bombieri-Lang conjecture.) We are ...
4
votes
0answers
531 views

Soft proof of multiplicity one for character groups of Shimura curves?

Is it not possible to prove mutiplicity one type statements for character groups of quaternionic Shimura curves by simply using Raynaud's description for character groups at primes dividing the ...
3
votes
2answers
229 views

spin bundle vs. hodge bundle

Let $\overline{\mathcal{M}}^r_{g,n}$ the space of $n$ pointed $r$-stable curves of genus $g$, endowed with a root of the canonical sheaf, and let $\mathcal{C} \to \overline{\mathcal{M}}^r_{g,n}$ be ...
9
votes
1answer
488 views

Can we always find a curve which doesn't have semi-stable reduction

Let $K$ be a number field and let $g$ be a positive integer. Does there exist a smooth projective geometrically connected curve $X/K$ of genus $g$ such that $X$ does not have semi-stable reduction ...
4
votes
2answers
655 views

The existence of meromorphic functions on Riemann surfaces

In Miranda's book on algebraic curves and Riemann surfaces, Miranda writes: It is a basic and highly nontrivial result that a compact Riemann surface has nonconstant meromorphic functions on ...
5
votes
5answers
837 views

The use of embedding a curve into its Jacobian

I'm looking for as many examples/applications as possible of the use of embedding a smooth projective geometrically connected curve $X$ over a number field $k$ with $X(k)\neq \emptyset$ into its ...
3
votes
2answers
113 views

Injectivity under flat base change of the Picard group on smooth projective curves

Let $K$ be a field of characteristic $0$, $X_K$ a smooth projective curve over $K$. Denote by $\bar{K}$ the algebraic closure of $K$. The base change morphism $X_{\bar{K}} \to X_K$, induces via the ...
2
votes
1answer
141 views

curve through a point avoiding an hypersurface, II

Inspired by this question: Suppose given an algebraic curve $C \subset \mathbb{A}^2$, and a point $x \in C$. Can you find another (closed) curve $D \subset \mathbb{A}^2$ such that $C \cap D = x$? ...
1
vote
0answers
124 views

Hopf lemma for line bundles on curves in algebraic geometry

In the paper http://arxiv.org/pdf/math/0110256v1.pdf Claire Voisin proves that all linear subspaces which lie inside of a (not too big) secant variety of a smooth projective curve must lie inside one ...