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Results tagged with ag.algebraic-geometry
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user 102390
Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology.
4
votes
If $R$ is an etale extension of $\mathbb Z$, then $R = \mathbb Z^n$?
Let $X$ be a normal integral scheme. Then $\pi_1^\mathrm{et}(X)$ is isomorphic to $\mathrm{Gal}(K(X)^\mathrm{un}/K(X))$, where $K(X)^\mathrm{un}$ is the compositum of all finite extensions $F$ of $K(X …
- 5,520
2
votes
Italian-style algebraic geometry in homotopy theory?
This is just a long comment. Homotopy theory is a rather broad field, so the answer to your question depends on what part of homotopy theory you'd like to see having interactions with "Italian-style" …
- 5,520
2
votes
1
answer
128
views
Pairs of quadratic forms and $\mathbf{A}^8/\mathrm{SL}_2^{\times 3}$
$\newcommand{\std}{\mathrm{std}}\newcommand{\SL}{\mathrm{SL}}\newcommand{\mmod}{/\!\!/}$Fix the base field to be the complex numbers $\mathbf{C}$. Let $\std = \mathbf{A}^2$ denote the standard represe …
- 5,520
5
votes
0
answers
262
views
Smooth morphisms to the moduli stack of elliptic curves
Fix a prime $p$, and let $\overline{M}$ be the Deligne-Mumford compactification of the moduli stack $M$ of elliptic curves (which, concretely, is the open substack of the stack of cubic curves which a …
- 5,520
8
votes
Accepted
Pairing of cotangent and tangent bundles
For (1), recall that if $R$ is a ring, then a derivation $D: R \to R$ satisfies the Leibniz rule, which by induction on $n$ implies that if $D^n$ denotes the $n$-fold iterate of $D$, then
$$D^n(fg) = …
- 5,520
3
votes
Twisted differential operator, chiral differential operator, $???$ (continue the sequence)
$\newcommand{\co}{\mathcal{O}} \newcommand{\H}{\mathrm{H}} \newcommand{\dR}{\mathrm{dR}} \newcommand{\GG}{\mathbf{G}}$
Let $k = \mathbf{C}$, let $X$ be a (smooth) variety over $k$, and let $X_\dR$ be …
- 5,520
5
votes
Is there a Hopf algebra-style description of chain complexes?
$\newcommand{\AA}{\mathbf{A}}\newcommand{\GG}{\mathbf{G}}\newcommand{\Mod}{\mathrm{Mod}}\newcommand{\fil}{\mathrm{fil}}\newcommand{\QCoh}{\mathrm{QCoh}} \newcommand{\spec}{\mathrm{Spec}}$Let $R$ be a …
- 5,520
4
votes
How to visualize local complete intersection morphisms?
As Dan Petersen said the comments, an lci morphism $f:X\to Y$ is precisely one that factors Zariski-locally as $X\to Y\times \mathbf{A}^n \to Y$, where $X\to Y\times \mathbf{A}^n$ is a regular embeddi …
- 5,520
6
votes
1
answer
504
views
Rationality of the moduli space of genus g curves
I'm not an expert in this topic, so please excuse my negligence. I'd also appreciate references to the literature. Throughout, I will work over the complex numbers, although the analogous questions co …
- 5,520
3
votes
Points of infinite level modular curve
The stack $M(p^n)$ of elliptic curves with a full level $p^n$-structure is canonically defined over $\mathbf{Z}_p[\zeta_{p^n}]$. Its generic fiber is the modular curve $X(p^n)$, and this determines a …
- 5,520
3
votes
Accepted
Homotopical interpretation of flatness?
The derived tensor product is defined as a homotopy pushout of $M\leftarrow R\rightarrow N$ in commutative DGAs. Now, if $R$ is a commutative ring and $M$ and $N$ are $R$-algebras, then the homology g …
- 5,520
5
votes
Any news about equivalences of periodic triangulated or $\infty$-categories?
$\newcommand{\QCoh}{\mathrm{QCoh}} \newcommand{\Mod}{\mathrm{Mod}} \newcommand{\LMod}{\mathrm{LMod}} \newcommand{\spec}{\mathrm{Spec}} \newcommand{\Fun}{\mathrm{Fun}} \newcommand{\Z}{\mathbf{Z}} \newc …
- 5,520
6
votes
Accepted
Spectral and derived deformations of schemes
In general, these are incredibly hard questions. It seems to me that one natural question to ask (if you are interested in $\pi_0$ of ring spectra) would be about understanding even periodic $\mathbf{ …
- 5,520
4
votes
Relationship between fans and root data
Not an answer, but: you can construct a fan from a root system. Let $R$ be a root system in an Euclidean space, and let $\Lambda_R$ be the root lattice with dual lattice $\Lambda_R^\vee$. The fan $\Si …
- 5,520