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1
vote
1answer
92 views

endomorphisms of the Jacobian of a curve

Let $C$ be a smooth, projective curve over the complex numbers and let $J(C)$ be its Jacobian. The Torelli theorem relates the automorphisms of $C$ to the automorphisms of $J(C)$. Precisely, ...
0
votes
1answer
102 views

Lipschitz boundary vs rectifiable curve boundary

I was looking at an old paper about domains with Lipschitz boundary. I am wondering, suppose that the boundary of a compact domain homeomorphic to a disk is a rectifiable injective curve : is this ...
10
votes
1answer
365 views

Number of curves over a finite field

Let $K$ be a finite field. Is there a formula for the number of isomorphism classes of genus $g$ smooth curves over $K$? In other words does there exists a formula for the number of rational points ...
0
votes
1answer
115 views

meaning of $k(C)/1+\mathfrak{m}_x$ [closed]

Let $C$ be a smooth projective curve over some field $k$ and $x$ a closed point of $C$. I've seen some constructions in which people use $k(C)^\times / 1+\mathfrak{m}_x$. What's the meaning of ...
3
votes
1answer
175 views

Etale covers of products of curves

Is a finite etale cover of a product of curves again a product of curves? The answer is no in general. Here's one way to construct an example. Take the product $A$ of two elliptic curves and an ...
2
votes
1answer
225 views

Kodaira dimension of the moduli space of curves

It is known that the moduli space $\overline{M}_{g}$ of genus $g$ curves is of general type for $g\geq 24$. By Theorem 2.4 of Logan, Adam The Kodaira dimension of moduli spaces of curves with ...
1
vote
1answer
144 views

Are Isom-schemes geometrically connected

This question is about properties of Isom-schemes that are well-known over algebraically closed fields. Let $K$ be a field of characteristic zero, let $C$ be a smooth projective geometrically ...
7
votes
1answer
217 views

Shimura surfaces that do not contain a Shimura curve

Let $S$ be a Shimura surface i.e. a Shimura variety with $dimS=2$. Does $S$ necessarily contain a Shimura curve? I know that probably the answer is No, but do not have an explicit example. What is the ...
3
votes
1answer
169 views

Structure of fundamental groups arising from smooth projective morphisms

Let $f:X\to B$ be a smooth projective morphism of complex algebraic varieties. If $f$ is of relative dimension zero, i.e., $f$ is a finite etale cover, then the image of the topological fundamental ...
0
votes
0answers
65 views

Dimension of the Representation of the Suzuki and Ree Groups?

What are the dimension of the group representation of $^2B_2$ and $^2G_2$? All what I know is that the first is 4 and the second group has two representation of dimension 7 and 13. Are there any?
2
votes
2answers
163 views

Relating the toric rank of a semistable curve and the first Betti number of its reduction graph

Introduction Let $k$ be a local field. Let $C$ be the spectrum of $\mathcal{O}_{k}$. Let $X/k$ be a smooth projective curve with a semistable model $\mathcal{X}/C$. Let $J$ be the Jacobian of $X$. ...
5
votes
2answers
371 views

radius of tubular neighborhood

Hi there, Is there any result about the calculation of radius of tubular neighborhood of submanifold inside a Riemannian manifold? For example, given a simple smooth curve on R^2, what's the radius ...
1
vote
0answers
176 views

Manipulating a Gaussian Distribution [closed]

Hello All, I warn in advance my math skills are quite limited so speaking to me like I am an idiot would be appreciated. I have a polynomial : $y = a\cdot e^{\frac{-2(x - x_0)^2}{w^2}}$ Now I know ...
5
votes
2answers
470 views

Is it possible to check two curves on birational equivalence by some computer algebra system?

I have two curves, for example hyperelliptic: \begin{align} &y^2 = x^6 + 14x^4 + 5x^3 + 14x^2 + 18, \\\\ &y^2 = x^6 + 14x^4 + 5x^3 + 14x^2 + 5x + 1 \end{align} Is it possible to check them ...
9
votes
4answers
921 views

Moduli space of genus 2 curves

Does any body know any reference in which the geometry of compactified moduli space of genus two curves ( Which is a three dimensional variety/stack/...) has been studied?