The connectedness of the moduli space M_g of complex algebraic curves of genus g can be proven by showing that it is dominated by a Hurwitz space of simply branched d-fold covers o …

Given a plane affine curve $\sum_{i,j}a_{i,j}X^iY^j = 0$, its genus can be calculated as the number of integral points of the interior of the convex hull of ${(i,j) \mid a_{i,j} \n …

I will state a very specific case: genus 5. Though it's particular, it admits a generalization to $M_g$, and I think reflects the nature of a general stratification of $M_g$. It i …

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 …

This is prop 2.6b on p.28 of Silverman's the Arithmetic of Elliptic curves. It says that let $\phi: C_1 \rightarrow C_2$ be a non-constant map of projective non-singular irreducib …

The Mumford conjecture states that for each integer $n$, we have: the map $\mathbb{Q}[x_1,x_2,\dots] \to H^\ast(M_g ; \mathbb{Q})$ sending $x_i$ to the kappa class $\kappa_i$, is a …

(I'm happy to work over an algebraically closed field....) Let $\mathcal{C} \rightarrow Spec (R)$ be a (flat) family of proper, prestable curves where $R$ is a DVR. Suppose the g …

Let $X$ be a smooth surface of genus $g$ and $S^nX$ its n-symmetrical product (that is, the quotient of $X \times ... \times X$ by the symmetric group $S_n$). There is a well know …

In going from Riemann surface theory to the theory of algebraic curves over fields $k$ that are not necessarily $\mathbb{C}$, I would like to understand more about how the notion o …

Is every curve over $\mathbf{C}$ birational to a smooth affine plane curve? Bonnie Huggins asked me this question back in 2003, but neither I nor the few people I passed it on to …

Here's something that's been bothering me, and that's come up again for me recently while reading some stuff about Hilbert schemes of points (Nakajima's lectures, specifically): L …

Suppose $\Gamma$ is a nice discrete subgroup of $SL(2,\mathbb{R})$ such that the genus of the Riemann surface $\mathbb{H}/\Gamma$ is larger than 1. We know that this Riemann surfac …

Let $X$ be a (edit: nonsingular) projective complex algebraic curve. The genus of $X$ can be defined as the dimension of the space of holomorphic $1$-forms on $X$, which in turn c …

The Torelli map $\tau\colon M_g \to A_g$ sends a curve C to its Jacobian (along with the canonical principal polarization associated to C); see this question for a description whic …

What is the best way to find an actual divisor of an affine curve? I.E. if I am interested in finding a canonical divisor of a curve in two variables, is there a general way to go …