Questions tagged [ag.algebraic-geometry]
for questions on algebraic geometry, including algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology.
16,086
questions
4
votes
0answers
63 views
Lifting a complete intersection in $\mathbb{P}^n_{\mathbb{F}_p}$ to $\mathbb{Z}_p$
Suppose that you are given a (not necessarily smooth) projective variety $X \subseteq \mathbb{P}^n_{\mathbb{F}_p}$ of codimension $d$ that is a complete intersection. In other words, it can be defined ...
1
vote
0answers
29 views
What is known about lower etale cohomology of unirational varieties?
Which information is currently known about $H^1_{et}(X,\mathbb{Z}_l)$ and $H^2_{et}(X,\mathbb{Z}_l)$, where $X$ is a smooth unirational variety over an algebraically closed field of finite ...
1
vote
0answers
45 views
formal smoothness and McQuillan formal schemes
Let $k$ be an algebraically closed field, $A\rightarrow B$ be a continuous map of weakly admissible topological $k$-local algebras.
We assume that it is formally smooth and topologically of finite ...
1
vote
0answers
33 views
Log canonical centers of toric (and toroidal) varieties
Q1: Let $(X,B)$ be a toric variety. There exists a toric resolution of singularities $f:(Y,E) \to (X,B)$. Here is my question:
Is any lc center of $(X,B)$ an irreducible component of an intersection ...
4
votes
0answers
84 views
Approximating ring maps of finite Tor-dimension
Let $R$ be a commutative ring, and let $S$ be a finitely presented $R$-algebra of finite Tor-dimension over $R$. Can $R \to S$ be realized as the base change, along some ring map $R_0 \to R$, of a ...
1
vote
0answers
38 views
Birational model of a log smooth pair
Given a log smooth pair $(X,B)$ with a reduced boundary divisor $B$, consider a birational model $\pi:X' \to X$ and a boundary divisor $B'$ which is given by $K_{X'}+B'=\pi^*(K_X+B)$. Here is my ...
2
votes
0answers
99 views
How exactly do we calculate an affine covering of a scheme?
Is there a general algorithm that takes in some scheme $X$ and outputs some affine covering of $X$ ? If not, how are affine coverings of schemes usually calculated ? I'm particularly interested in ...
11
votes
1answer
227 views
Calabi-Yau threefold with an automorphism of infinite order
I am looking for a (hopefully simple) example of a Calabi-Yau threefold (projective, simply connected, with trivial canonical bundle) admitting an automorphism of infinite order.
4
votes
0answers
224 views
Modern context for hypercohomology spectra
In Thomason's paper Algebraic K-theory and étale cohomology, (Ann. ENS 1985, Numdam link) Thomason develops an elaborate theory of hypercohomology spectra, $\mathbb{H}(X,\mathcal{F})$ for presheafs of ...
1
vote
1answer
93 views
Is the set of images of an open subset of full-rank matrices an open subset of the Grassmannian?
$\DeclareMathOperator\Gr{Gr}$ Let $\Gr(k,n)$ be the set of $k$-dimensional subspaces in affine space $\mathbb{A}^n$ over an algebraically closed field. If $U\subseteq (\mathbb{A}^n)^{\times k}$ is an ...
1
vote
2answers
166 views
Faithfully flat modules over a group algebra
Suppose we have the following data:
1) A group ring $\mathbb{Z}[G]$, where $G$ is a torsion free group.
2) $M_{\bullet}$ a bounded (above and below) chain complex of $\mathbb{Z}[G]$-modules such ...
0
votes
1answer
96 views
Independent conditions imposed by points in different planes
Let $H_1$ and $H_2$ be two planes in $\mathbb{P}^3$.Let $P$ be a set of $9$ points such that no three lie on a line. Suppose $H_1$ contains 4 of them and $H_2$ contains remaining $5$ points. Is it ...
4
votes
0answers
83 views
Reference request - conjugacy classes over local fields
Is there a nice reference for reductive groups over local fields, which for example contains discussion of things such as: Given a semisimple element in $G(F)$, its $G(F)$-conjugacy class is closed in ...
-1
votes
0answers
54 views
Determing a pattern to a set of quadruples [closed]
Here are three families of quadruples $(a, b, c, d)$. Each quadruple has the property
that $a^3+b^3+c^3=d^3$.
Each family is generated by a certain rule. What is a pattern
that generates them, and ...
0
votes
1answer
76 views
IntersectInP bug of Macaulay2 [closed]
I am trying to use the intersectInP command in Macaulay2, inside package ReesAlgebra. However, I tried to follow the exact code in the user-guide, but it doesn't run in my Ubuntu app (of win 10). Can ...
1
vote
1answer
184 views
Viewing a finite group as a group scheme
I've read books which have this statement, without explanation : 'every finite group is an algebraic group'. I'm trying to understand what exactly they mean. The definition I have in my mind of a ...
3
votes
0answers
123 views
Algebraic de Rham cohomology for singular varieties
I would like to know to what extent the naive algebraic de Rham cohomology is a "bad" cohomology theory. If $X$ is smooth then there is a comparison theorem with singular cohomology. If $X$ is ...
2
votes
0answers
61 views
Set of orthogonal complements to open set in $Gr(k,\mathbb{C}^n)$ open in $Gr(n-k,\mathbb{C}^n)$?
$\DeclareMathOperator\Gr{Gr}$Consider $\mathbb{C}^n$ endowed with the Hermitian inner product $\langle u,v\rangle=u^*v$, and let $U \subseteq \Gr(k,\mathbb{C}^n)$ be a Zariski open dense subset of the ...
7
votes
0answers
70 views
+50
Resolution graphs in the sense of Nemethi
The following definitions are from lecture notes of Nemethi:
A surface singularity $(X,0)$ is defined by $$(X,0) = (\{ f_1 = \ldots = f_m=0 \}) \subset \mathbb (C^n,0),$$ where $f_i : (\mathbb C^n ,0)...
1
vote
1answer
82 views
complete intersection curves with large Hilbert scheme of points
Let $X$ be a very general hypersurface of degree $6$ in $\mathbb{P}^3$. Fix an integer $d$.
Define $Y:= \{ C \in \mathbb{P}(H^0(\mathcal{O}(3))) \text{ such that } \text{dim}(\text{ Hilb}^d(X \cap C)) ...
2
votes
1answer
174 views
(1/2) K3 surface or half-K3 surface: Ways to think about it?
I heard from string theorists thinking of the so-called "(1/2) K3 surface" or "half-K3 surface" as follows:
Let $T^2 \times S^1$ be a 3-torus with spin structure periodic in all directions. $T^2 \...
1
vote
1answer
142 views
Can the degree of an affine variety increase after intersecting with a hyperplane?
Say we have an affine variety $V \subset \mathbb{C}^n$, and suppose we intersect $V$ with a hyperplane $H$, possibly not in general position. Is it possible for the degree of $V \cap H$ to be larger ...
0
votes
1answer
91 views
basic question on quotient stacks
Let $X$ be a scheme over $S$, and $G$ be an affine group scheme over $S$ acting on $X$. This Wikipedia article (or also this related MO question) defines a quotient stack $[X/G]$ as a category of ...
2
votes
0answers
164 views
The connectedness of the coarse moduli scheme of $\mathscr{M}_*(N)$ over $\mathbb{Z}$
Let $N \ge 1$ be an integer and $\mathscr{M}_*(N)$ the stack of elliptic curves with the level $\Gamma(N), \Gamma_0(N), \Gamma_1(N)$ or $\Gamma_{\text{bal.}1}(N)$.
(For its definition see Katz-Mazur....
1
vote
0answers
36 views
powers of linear forms in projections of complete intersections in codimension 3
Let $I\subset \mathbb{C}[x_0,x_1,x_2]=:A$ be a complete intersection, $I=(p_1,p_2,p_3)$, $p_i$ homogeneous all of the same degree d
for some $d>2$.
Let $l$ be a general linear form and let $J$ ...
2
votes
0answers
100 views
Reference request: Singular curves
I'm interested in coherent sheaves on a singular curve.(For example, global dimension, Serre duality, Riemann-Roch's theorem for singular curves,etc....)
I find treatment of it only in Hartshorn's ...
3
votes
0answers
77 views
Special irreducible polynomials in $k[x,y]$
The following question I have asked in MSE, getting one comment.
Hopefully, it is ok to ask it here also.
Let $k$ be a field of characteristic zero, $n \in \mathbb{N}$.
Definitions:
(1) $0 \neq f \...
55
votes
6answers
9k views
Why is the Fourier transform so ubiquitous?
Many operations and equivalences in mathematics arise as some sort of Fourier transform. By Fourier transform I mean the following:
Let $X$ and $Y$ be two objects of some category with products, and ...
3
votes
1answer
175 views
Locally free group scheme étale
Let $R$ be a commutative ring, $p >0$ prime and $G$ a finite, locally free group scheme over $R$ of rank $p^n$; $n \in \mathbb{N}_{\ge 1}$. Assume $p \in R^*$ (i.e. is a unit in $R$).
Question: ...
3
votes
0answers
77 views
Multiplicative structure of the K-theory of Severi-Brauer varieties
There is a well-known result by Quillen stating that if $X_A$ is the Severi-Brauer variety of a central simple algebra $A$ of degree $d$ over a field $k$, then its (Quillen) K-theory decomposes as
$$...
9
votes
0answers
125 views
Good reduction of finite etale covers of abelian varieties
Let $R$ be a dvr (whose residue characteristic is zero if it helps) with fraction field $K$.
Let $A$ be an abelian variety over $K$ with good reduction over $R$. Let $X\to A$ be a finite etale ...
0
votes
0answers
108 views
Neukirch's theorem on absolute Galois groups in English [duplicate]
Is there a paper or book available in English that proves the result of Neukirch on absolute Galois groups of number fields? I'm having a hell of a time with the German originals.
11
votes
4answers
830 views
Book on manifolds from a sheaf-theoretic/locally ringed space PoV
I'm looking for an introductory (or rather, non-advanced) book on manifolds as locally ringed spaces, i.e., from the algebraic geometric viewpoint. Most introductory texts only introduce manifolds ...
0
votes
0answers
27 views
Extension-closed subcategory $P(I)$ defined by stability condition $(Z, P)$ and an interval $I \subset \mathbb{R}$
Let $D$ be a triangulated category, and let $\sigma = (Z, P)$ be a Bridgeland stability condition on $D$. Let $I \subset \mathbb{R}$ be any interval (open, closed, or half-open).
The category $P(I)$ ...
3
votes
0answers
180 views
What is the left adjoint to base change of schemes?
Restriction of Scalars and Functoriality of Presheaves.
Let $\phi\colon R\longrightarrow S$ be a morphism of rings. There is associated to $\phi$ a natural functor from $\mathrm{Alg}_S$ to $\mathrm{...
0
votes
0answers
95 views
Integral closure of graded rings
Let $S = \bigoplus_{d\ge0} S_d$ be a graded ring which is an integral domain. Is the integral closure of $S$ in its quotient field also graded?
2
votes
0answers
25 views
grobner basis of an ideal dependent on some parameter
Suppose $I = \langle f_1, ... , f_l \rangle$ is an ideal generated by polynomials $f \in k[x_1,\dots,x_n]$, where $k$ is a field of rational functions in some parameters $s_1,\dots,s_m$.
What are the ...
4
votes
0answers
86 views
program to compute hurwitz numbers
Is there a computer program available to compute Hurwitz numbers easily? In fact I only care about counting covers $C\to\mathbb{P}^1$ branched over $0,1,\infty$, and am even willing to restrict to the ...
8
votes
0answers
311 views
+100
Hard: One more generator needed for a Z/6 elliptic curve
We are searching for rank 8 elliptic curves with the torsion subgroup $\mathbb{Z}/6$ using newly discovered families similar to Kihara's as described in
A. Dujella, J.C. Peral, P. Tadić, Elliptic ...
8
votes
1answer
296 views
Homomorphism induced by the second exterior power of a linear map
Consider the map from $M(n, \mathbb Z) \rightarrow M(\binom{n}{2}, \mathbb Z)$ taking a matrix A to its second compound, i.e, $\bigwedge^2 A$.
Restricting this map to the invertible matrices we get a ...
3
votes
0answers
142 views
Functorial interpretation of formal completion
Let $S$ be a scheme, $X/S$ a "nice" scheme (I think "nice" = separated) and $e : S \to X$ a section.
Let $\hat{X}$ be the formal completion of $X$ along the section $e$.
(i.e., = $\lim_n S_n$, where $...
2
votes
0answers
110 views
Confusion about the classification of isotrivial group schemes
in SGA 3, exposé X, we find the following classification result (corollaire 1.2):
Let $S$ be a connected scheme and let $\xi:\mathrm{Spec}(\Omega) \to S$ be a geometric point. Let $\pi=\pi_1(S,\xi)$ ...
2
votes
0answers
117 views
Stalks of perverse cohomology sheaves?
For a complex of sheaves $\cal{F}^{\bullet}$ on a variety $X$, a useful fact is that the stalks of the cohomology sheaves of $\mathcal{F}^{\bullet}$ agree with the cohomology groups of the complex of ...
4
votes
0answers
204 views
+100
Artin's “On Isolated Rational Singularities of Surfaces”
My question refers to M. Artin's paper "On Isolated Rational Singularities of Surfaces"; more precisely the proof of Thm 4 on page 133. Here the relevant excerpt:
The Setting: Let $\bar{V}=Spec(A)$ ...
4
votes
0answers
73 views
Are there any explicit (prime-to-l) alterations for interesting varieties (or schemes)?
I have read that it is easier to find regular alterations of varieties than their resolutions of singularities (moreover, I believed in this sentence when I read it). My question is: do there exist ...
11
votes
1answer
320 views
Factorization and vertex algebra cohomology
A chiral algebra on a smooth curve $X$, in the sense of Beilinson-Drinfeld, is a right $D_{X}$-module with a chiral bracket, which is a map $\mathcal{V}^{\boxtimes 2}(\infty\Delta)\rightarrow \Delta_{*...
3
votes
2answers
238 views
Does local cohomology commute with pullback?
Let $Y$ be a topological spaces and $Z \subset Y$ be locally closed, i.e. $Z=V \cap U^c$ for $U,V \subset Y$ open.
For any abelian sheaf $\mathcal{F}$ on $Y$ let $\Gamma_Z(Y,\mathcal{F}):=\ker(\...
2
votes
0answers
101 views
Density of rational points over finite fields
Let $X\hookrightarrow\mathbb P^n_{\mathbb F_q}$ be a geometrically integral hypersurface over the finite field $\mathbb F_q$ of degree $\delta$. In order to estimate the number of its rational points, ...
4
votes
0answers
94 views
Name for “étale-essential” properties
A map of rings $f:A\to B$ is called "essentially $P$" if there exists some $A\to C\to B$ such that $A\to C$ has property $P$ and $C\to B$ is a localization, that is to say, a filtered colimit of ...
3
votes
0answers
149 views
Relations between rational algebraic K-theory and Chow groups
A consequence of Grothendieck's Riemann-Roch Theorem is the fact that the Chern character induces an isomorphism between algebraic
$ch: K_{0}(X) \otimes \mathbb{Q} \stackrel{\cong}{\rightarrow} C H^{*...
