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This question comes (heavily edited) from my notes, thus slightly unusual structure. We know that algebraic maps have very strict structure, and in many settings the operations f_*, f_!, their adjo …

Here's an example paper: 0905.2212 It uses some bound on Castelnuovo–Mumford regularity to prove that cohomology of smooth complex projective variety can be computed in parallel polynomial time.

If you have a line bundle trivial on 3 "surfaces" of a "cube" $A\times B\times C$ where $A$, $B$, $C$ are abelian varieties, then this line bundle in trivial on the whole "cube". See wikipedia.

One thing I understand is that vanishing cycles are more than just about singularities — there's a derived version that is more interesting. I'd like to get an answer myself. This is also important fo …

Update: at the bottom there's a wonderful and fresh reference. There's no field with one element in the literal sense, but there are constructions that work over different fields $\mathbb F_q$ and wh …

First of the questions about derived algebraic geometry from a noobie. The way I understand it, every discrete ring $R$ corresponds to some ring spectrum whose $\pi_0$ is $R$. Now consider $p$-adic nu …

Motivation I learned about this question from a wonderful article Rational points on curves by Henri Darmon. He gives a list of statements (some are theorems, some conjectures) of the form the set $\ …

From a post to The Jouanolou trick: Are all topologically trivial (contractible) complex algebraic varieties necessarily affine? Are there examples of those not birationally equivalent to an affin …

I think I'll be collecting references I found in this answer, rather then in the original (already large) post: http://www-fourier.ujf-grenoble.fr/~peters/hodge.f/peters-proc.pdf (Wayback Machine) I …

Here's a problem that may ultimately require just simple algebraic-geometry skills to be solved, or perhaps it's very deep and will never be solved at all. From the comments, some literature and my me …

There was an observation that any algebraic curve over Q can be rationally mapped to P^1 without three points and this led Grothendieck to define a special class of these mappings, called the Children …

The book of Kashiwara and Schapira is quite focused and technical. I won't recommend it as an introduction to sheaves, since the abstract language of sheaves and homological algebra is most useful whe …

Suppose for some reason one would be expecting a formula of the kind $$\mathop{\text{ch}}(f_!\mathcal F)\ =\ f_*(\mathop{\text{ch}}(\mathcal F)\cdot t_f)$$ valid in $H^*(Y)$ where $f:X\to Y$ is a …

There are some things about geometry that show why a motivic viewpoint is deep and important. A good indication is that Grothendieck and others had to invent some important and new algebraico-geometri …

is there a way to glue two schemes together along a closed point (say we're working over a field)? Is it easier to glue in the category of algebraic spaces? For this particular pushout, the geome …