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We consider the variety $\Sigma_{m}$ := {($p$, $X$) : $X$ is a degree $n$ + 1 hypersurface over $\mathbb{C}$ with mult$_{p}(X) \geq m$} $\subseteq$ $\mathbb{P}$$^{n}$ $\times$ $\mathbb{P}$$^{N}$, ...

I'm reading Mumford's book Geometric Invariant Theory and confused about the proof of a lemma on Page 91&92: Lemma: Let $V_0$ be a smooth surface over an algebraically closed field $k$ with char$...

Let $\mathsf{k}$ be a field of characteristic $0$, and consider $\mathsf{k}(x,y)$. If $\mathsf{k}$ is algebraically closed, then every field $L$ such that the inclusion $\mathsf{k} \subset L \subset \...

I recently worked through most of the proof of the rationality of the moduli of genus 3 curves, which seemed to have the following structure: Every nonhyperelliptic genus 3 curve is a smooth plane ...