algebraic curves - Determining numerical class of divisors inside Jacobian

Let $C$ be a smooth projective curve of genus $g$, let $c$ be a point in $C$. Let $(n_1,\dots, n_{g-1})$ be a $(g-1)$ tuple of nonzero integers. Consider the image $f_{(n_i)}\colon C^{g-1}\to \mathrm{Pic}^0(C)$ given by $$(x_i)\mapsto \mathcal{O}_C\left(\sum n_i(x_i-c)\right),$$the image $\mathrm{Im}(f_{(n_i)})$ is a divisor in $\mathrm{Pic}^0(C)$. If $C$ is general, then $NS(C)$ is generated by theta divisor $\Theta$, so we can write numerical classes as a multiple of the theta divisor,$$\mathrm{Im}(f_{(n_i)})=d(n_1,\dots,n_{g-1})\Theta.$$Is ...Read more

ag.algebraic geometry - Families of curves for which the Belyi degree can be easily bounded

I know (edit: three) families of smooth projective connected curves over $\bar{\mathbf{Q}}$ for which the Belyi degree is not hard to bound from above. The modular curves $X(n)$. They are constructed by compactifying the quotient $Y(n) = \Gamma(n)\backslash \mathbf{H}$. The natural morphism $X(n) \longrightarrow X(1)$ is Belyi of degree $n^2$ (up to a constant factor). This also bounds the Belyi degree of a modular curve given by a congruence subgroup $\Gamma$. In general, Zograf proves that the Belyi degree of a (classical congruence) modular ...Read more

algebraic curves - Mapping class group of a punctured genus 0 surface

Let $T_{0,n}$ be the Teichmuller space of $n$-punctured genus $0$ Riemann surface, and $M_{0,n}$ the Moduli space (assume $n\geq 3$ and the punctures are numbered). What is the correct notion of the mapping class group $\Gamma_{0,n}$, so that $M_{0,n} \cong T_{0,n} / \Gamma_{0,n}$? Now $\pi_1(M_{0,n}) \cong \Gamma_{0,n}$ since the Teichmuller space is contractible and the action of $\Gamma_{0,n}$ is free.What does the Dehn-Nielsen-Baer theorem say in this case? Let $S_{0,n}$ be the $n$ punctured complex projective line then which subgroup of $O...Read more

Motivation for Zeta Function of an Algebraic Variety

If $p$ is a prime then the zeta function for an algebraic curve $V$ over $\mathbb{F}_p$ is defined to be$$\zeta_{V,p}(s) := \exp\left(\sum_{m\geq 1} \frac{N_m}{m}(p^{-s})^m\right). $$where $N_m$ is the number of points over $\mathbb{F}_{p^m}$.I was wondering what is the motivation for this definition. The sum in the exponent is vaguely logarithmic. So maybe that explains the exponential?What sort of information is the zeta function meant to encode and how does it do it? Also, how does this end up being a rational function?...Read more

algebraic curves - Units of Endomorphism Rings of Jacobian Varieties with Real Multiplication

Let $(A,a)$ be a principally polarised (with indecomposable polarisation) Abelian variety over $\mathbb C$. Assume that End(A) contains an order $R$ of a totally real number field of degree $>1$ over $\mathbb Q$ and that $R$ respects the polarisation of $A$. The units of $R^*$ are therefore automorphisms of $(A,a)$.If $g = \dim(A)$ and $g = \text{rank}_{\mathbb Z}(R)$ then my understanding is that such $(A,a)$ are parameterised by a Hilbert Modular variety of dimension $g$. Therefore if $g$ is small, the Hilbert modular variety will inter...Read more

ag.algebraic geometry - Which curves have stable Faltings height greater or equal to 1

Let $Y$ be a smooth projective connected curve of genus $g>0$ over $\overline{\mathbf{Q}}$. Let $h_{\textrm{Fal}}(Y)$ be the Faltings height of $Y$. Question 1. Can one classify or describe the curves $Y$ such that $h_{\textrm{Fal}}(Y) \geq 1$?Question 2. For any $g>0$, does there exist a curve $Y$ of genus $g$ such that $h_{\textrm{Fal}}(Y) <1$? Essentially, I would like to know which curves one is excluding by looking at curves $Y$ such that $h_{\textrm{Fal}}(Y) \geq 1$. A result of Bost says that the stable Faltings height of an ab...Read more

Disjoint curves in an algebraic surface

Let $X$ be an algebraic surface (over the complex) with $p_g=q=0$. Is it possible to have disjoint curves $C_1,\ldots, C_b$, of positive genus, spanning $H_2(X,{\mathbb Q})$, $b=b_2(X)$?(When $X$ is the blow-up of the projective plane at some points, the exceptional divisors and a curve coming from the projective plane not passing through the points, give an answer if we do not assume the condition on the genus.)...Read more

riemann surfaces - Double Cover Map of Hyperelliptic Curve is Unique

On page 204 of Rick Miranda's Algebraic Curves and Riemann Surfaces, he talks about how the canonical map of a hyperelliptic curve is the double cover map composed with the Veronese map $\phi(x) = [1:x:\cdots:x^{g-1}]$. Then, he says:the double covering map for a hyperelliptic curve of genus $g\ge 2$ is unique since is it the canonical map after all. I don't understand what is meant by unique and how this follows from the discussion....Read more

algebraic curves - Prove that $H^1(\mathcal{M}^*)=0$.

Let $X$ be a compact Riemann surface. For an open set $U$, let $\mathcal{M}^*(U)$ be the multiplicative group of nonzero meromorphic functions on $U$ ("nonzero" meaning "not identically zero"). This makes a sheaf $\mathcal{M}^*$. Proposition: $H^1(\mathcal{M}^*)=0$.Miranda remarks this, claiming that $\mathcal{M}^*$ is a constant sheaf and the proposition follows.A sheaf $\mathcal{F}$ is constant iff there is a group $A$ and an open cover $\{U_i\}$ such that $\mathcal{F}(U_i) = A$ for all $i$, right? Is it equivalent to say that $\mathcal{F}_p...Read more

moduli spaces - Trigonal curves of genus three: can their Galois closure be non-abelian

Let $X$ be a curve of genus three which is not hyperelliptic. Then $X$ is trigonal, i.e., there exists a finite morphism $X \to \mathbf P^1$ of degree $3$.Let $Y\to X \to \mathbf P^1$ be a Galois closure of a trigonal curve. Then, unless $X\to \mathbf P^1$ is the Klein curve, the map $Y\to \mathbf P^1$ is of degree $6$. Now, surely the latter map could have non-abelian Galois group, but I don't know any explicit examples. Can somebody give me an explicit example for which it is clear that the Galois group over $\mathbf P^1$ of the Galois closur...Read more

abelian varieties - Abel-Jacobi map for Mumford curves analytically

Let $K$ be a field equipped with a non-Archimedean absolute value, let $\Gamma$ be a Schottky group in $PGL_2(K)$, and let $X_\Gamma$ be the associated Mumford curve, which is a proper smooth rigid analytic space over $K$, but also a proper algebraic curve (due to Mumford's theorem). Let $J_\Gamma$ be the Jacobian variety of degree 0 divisors on $X_\Gamma$, and let $a: X_\Gamma \to J_\Gamma$ be the Abel-Jacobi map which sends a point $x$ to the class of the divisor $[x-x_0]$ for some $x_0 \in X_\Gamma$ fixed in advance. It is known that $J_\Gam...Read more

abelian varieties - Geometric (or at least non-cohomological) proof of Lefschetz trace formula for curves

There is an isomorphism between (rational) correspondences on a curve $C/\mathbb{F}_p$ orthogonal to the "valence zero" ones (i.e. orthogonal under intersection pairing to $\{*\}\times C$ and $C\times \{*\}$) and endomorphisms of the Jacobian $J(C)$, giving a pretty and applicable demonstration of how $J(C)$ "is" the motivic weight-$1$ part of the curve, and illustrating the simplest case of the Lefschetz trace formula. One can show that this is an isomorphism in a pretty nice geometric way (from an endomorphism, get a correspondence by looking...Read more

Trefoil knot as an algebraic curve

Is the trefoil knot with its usual embedding into affine $3$-spacean algebraic curve (maybe after extending scalars to $\mathbb{C}$)? Is there even some thickening to some algebraic surface? If not, is there at least some similar algebraic curve which describes this type of knot? I hope that this question is not silly, I know almost nothing about this classical stuff on algebraic curves. A google research indicates that there is some connection with the cusp $y^2=x^3$, but I don't really get it.PS: I am interested in explicit equations. Specifi...Read more