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Once in a while I run into literature that invokes vanishing cycle machinery with a cryptic sentence like, "this follows from a standard vanishing cycle argument." Is there a good way to look at vani …

The basic answer is essentially as Emerton described in the comment. The most commonly used topologies on schemes are Zariski, Nisnevich, étale, smooth, syntomic, fppf, and fpqc, and this list is tot …

If $k$ is a characteristic $p$ field containing a subfield with $p^2$ elements (e.g., an algebraic closure of $\mathbb{F}_p$), then the number of isomorphism classes of supersingular elliptic curves o …

Vote this answer up if you consider yourself an algebraic geometer, and (in the course of your work) actually see nice pictures of lines, surfaces, and curves in your head.

Last fall, there was a conference in Nagoya about precisely this question (oddly enough, funded by a "Riemann Hypothesis" DARPA grant). Since I was attending a different conference at the same univer …

My experience with foundations is rather scant, but I've run into some types of objects that seem to resist the sort of set-theoretic encoding schemes via Kurowski tuples that are rather common for ob …

Yes. See Kedlaya's Fourier transforms and p-adic Weil II. This is a proof using Berthelot's rigid cohomology.

If you're just looking for a quick overview, you may want to read the lecture notes from the 2009 MIT K-theory lunch seminar, especially the first five lectures

The Max-Cut problem for a graph asks for a subset $S$ of vertices such that the number of edges between $S$ and the complement of $S$ is maximized. This problem is NP-hard. In fact, Håstad showed th …

I can't really say that I understand what a weight is, but the qualitative distinction between weight zero and positive weight has come up a couple times in MathOverflow questions: The étale fundame …

One way to think of twisted $D$-modules that I like is to view them as monodromic $D$-modules (see Beilinson, Bernstein A Proof of Jantzen Conjectures section 2.5, available as number 49 on Bernstein' …

Any abstract algebraic compact complex manifold is Moishezon. By Moishezon's theorem, any Kähler Moishezon manifold is projective algebraic. There are non-projective proper complex varieties, so $X_ …

I suggest looking at the illustrations in The Geometry of Schemes by Eisenbud and Harris and Algebraic Geometry by Hartshorne. Both books have many carefully drawn pictures of schemes that are not va …

The category of maps from a test object $T$ to a quotient stack $[X/G]$ has the following general form. Objects are pairs $(P, f)$, where $P$ is a $G$-torsor over $T$, and $f: P \to X$ is a $G$-equiv …

A bit of mastication of Katz-Mazur Theorem 13.7.6 and the surrounding text seems to yield the following description of the special fiber of $Y(p)$: It is fundamentally $p+1$ copies of $\mathbb{P}^1$ …