algebraic geometry Archives - Page 2 of 2

A proper scheme with infinite-dimensional fppf cohomology

In algebraic geometry, very often one encounters theorems of the following flavor:

Theorem: Let $f : X \to S$ be a proper morphism of spaces. Then for every sheaf $\mathcal{F}$ on $X$ that is finite, so is its pushforward $Rf_\ast \mathcal{F}$ .

Notice how I was being deliberately vague in the theorem above. What are $X$ and $Y$ ? What does …

Looking for a better counterexample

This post is doubling my old question asked on Mathoverflow.

Let $X$ be a smooth projective complex variety of dimension $d$ , and let $Y\subset X$ be an ample irreducible divisor. Recall that a cohomology class $b\in H^k(X,\mathbf{Q})$ is called $Y$ -primitive, if $b\smile c_1(Y)^{d-k+1} =0$ . Here, $c_1(Y)\in H^{2}(X,\mathbf{Q})$ is the first Chern class of the line bundle associated to $Y$ . …

The étale cohomology of curves over finite fields

When I was a graduate student, Zev Rosengarten (a former student of Brian Conrad) and I used to eat dinner at Stanford’s Arrillaga dining hall a lot. We’d talk about math for hours, but one thing that will forever be ingrained in my mind is how Zev was able to …

A shortcut in Kapovich’s proof of Haupt’s theorem

The Teichmüller space $T_g$ of genus $g$ curves carries the Hodge bundle $\Omega T_g$ , the total space of which maps into the first cohomology space $V = H^1(S_g,\mathbb{C})$ via the period map (i. e., a holomorphic 1-form maps into its cohomology class). Haupt’s (or Haupt–Kapovich) theorem describes the image $per(\Omega T_g) \subset V$ in terms of the integral structure on $V = H^1(S, \mathbb{Z}) \otimes \mathbb{C}$ and …