" Every curve of the n-th order is in a flat space of n dimensions or less " and " If there be a system of $n + m + 1$ quantities $x$ connected by $n + m - 1$ homogeneous equati …

If $f: \mathcal{C} \rightarrow \mathcal{P}_{2g+2}$ is a general family of hyperelliptic curves (defined over $\mathbb{C}$), we know that the algebraic connected monodromy group $M …

Fix a number field $K$. Is the rank of $J(K)$ unbounded, where $J$ ranges over the Jacobians of all smooth, projective, geometrically connected curves over $K$? Does there exist …

Hi, Are the Igusa invariants (defined in Igusa's oringal paper) also classical Siegel modular forms? I read from somewhere that $\psi_4=\frac{1}{4}I_4, \quad \psi_6=\frac{1}{8}(I …

1) To parameterize the conic $x^2+y^2=1$, we can use $(x,y)=(\sin t,\sin't)$ ($\sin'$ meaning the derivative of $\sin$, namely $\cos$). 2) To parameterize an elliptic curve $y^2=4 …

Let's consider spatial algebraic curve $C\subset \mathbb P^3$. How could I describe a family of such curves, for example the set of all curves genus $g$ passing through $k$ points? …

I am not sure if this is a valid question but posting any way: Say I am over $\mathbb{F}_{p}$ for a prime $p$. I have a curve of form $x^{2} = f(y)$ where $f(y)$ has an unknown f …

Genus 0 curves are well understood in number theory. There is also are rich theory a bunch of conjectures about the arithmetic of elliptic curves. This leads me to the question, w …

This question is based on Beauville's article in Szpiro's asterisque Seminaire sur les pinceaux de courbes de genre au moins deux from 1986. We will work over the complex numbers …

We will work over the complex numbers C. This question is based on Beauville's article : there exist a non-isotrivial fibration of genus 2 over P^1 with only 3 singular fibres. …

We will work over the complex numbers C. there exist a fibration of genus two over P^1 with only 3 singular fibres but one is semi-stable fiber. there not exist a fibration of …