4,178 reputation
1229
bio website math.berkeley.edu/~achinger
location Berkeley
age 28
visits member for 4 years, 10 months
seen 5 hours ago
PhD student, UC Berkeley

Dec
9
comment Morphism on schemes induced by continuous morphism of sites
Do you mean ringed topoi?
Dec
5
comment Rationality of moduli spaces of rational curves
@AbdelmalekAbdesselam good point! :D
Dec
4
comment Rationality of moduli spaces of rational curves
Maybe I'm wrong, but doesn't the following work? For $a_0, \ldots, a_{n-1}\in k$, consider the subscheme of $\mathbb{A}^1_k$ given by $x^n + a_{n-1}x^{n-1} + \ldots + a_0 = 0$. For generic $a_i$, this is a collection of $n$ distinct points, so we get a rational map from $\mathbb{A}^n$ to $\tilde M_{0, n}$ which is actually birational (as it's dominant and injective).
Dec
3
comment Map of adjunctions
For me (an algebraic geometer), the natural setting would be a cartesian diagram of topoi, and in that case we would call the natural transformation the "base change" map. There are important cases where this map is an isomorphism, e.g. when the map $H$ is "proper" (this is the $q=0$ part of the proper base change theorem in SGA4). So I would try to characterize properness in category-theoretic terms.
Dec
1
awarded  Good Answer
Nov
26
comment How to see that this pairing of line bundles is multiplicative?
But $F$ might not have a finite resolution by vector bundles unless $X$ is smooth projective...
Nov
25
comment How to see that this pairing of line bundles is multiplicative?
What is the definition of $\det(F)$? $F$ might not be a perfect complex...
Nov
25
awarded  Enlightened
Nov
25
awarded  Nice Answer
Nov
25
comment Etale Slice Theorem
Think of the case where the action is free. Then the theorem says that $X$ is a $G$-torsor over $X/G$.
Nov
25
comment Massive cancellations
Yes, I agree it should be true for sets of algebraic numbers, but it seems like a very difficult problem in diophantine approximation. Can you deal with the case $A=\{a, b\}$ with $a$ rational and $b$ algebraic?
Nov
25
comment Massive cancellations
Indeed, my $A$ consists of two elements.
Nov
25
answered Massive cancellations
Nov
10
revised On the infinitesimal lifting property of non-singular affine schemes
edited body
Nov
10
answered On the infinitesimal lifting property of non-singular affine schemes
Nov
7
comment Obtaining non-normal varieties by pushout
I think the claim above could follow from some results in Artin's paper "Algebraization of Formal Moduli II", where instead of $Z_n$ we consider their limit (the formal completion). The trick seems to be to choose an appropriate $n\gg 0$.
Nov
7
comment Obtaining non-normal varieties by pushout
Just a comment. One might hope for the following: given a variety $X$ and a proper birational $f:X'\to X$, let $Z\subseteq X$ be the locus where $f$ is not a local isomorphism (with reduced subscheme structure). For $n\geq 1$, let $Z_n$ denote the $n$-th infinitesimal thickening of $Z$ in $X$. Then for $n\gg 0$, $X$ is the pushout of $Z_n \leftarrow f^{-1}(Z_n)\to X'$. To answer your question, we would apply this to the normalization map (as hinted by Karl Schwede).
Oct
12
awarded  Benefactor
Oct
12
accepted Can we normalize a complex analytic space in a covering of an open subset?
Oct
5
awarded  Promoter