Questions tagged [ag.algebraic-geometry]
Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology.
21,608
questions
4
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0
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87
views
Is $x \in A_1$ left algebraic over the subalgebra generated by $p$ and $q$, $[q,p]=1$?
Let $A_1:=A_1(x,y,k)$ be the first Weyl algebra over a field $k$ of characteristic zero,
namely, the $k$-algebra generated by $x$ and $y$ with relation $yx-xy=1$.
Let $f:(x,y) \mapsto (p,q)$ be a $k$-...
- 2,783
3
votes
1
answer
234
views
Is the quotient of resolution the same as resolution of the quotient?
Suppose $X$ is a singular variety over a field $k$, which admits an action by a finite group $G$. Suppose the quotient $X/G$ is also a variety over $k$. If $Y_G$ is a resolution of $X/G$, does there ...
- 2,961
7
votes
2
answers
425
views
Unirationality of Fermat varieties in characteristic $p$
In Shioda's famous paper "An Example of Unirational Surfaces in Characteristic $p$" (MSN), the author proved that the Fermat surface over the characteristic-$p$ field $k$
$$
x_1^n +x_2^n + x_3^n+x_4^...
- 191
15
votes
2
answers
1k
views
Axiomatic characterization of virtual fundamental classes?
There are several objects called "virtual fundamental classes." For example, certain Deligne-Mumford stacks, quasi-smooth derived schemes, etc. will admit a "perfect obstruction theory" as defined by ...
user100272
14
votes
0
answers
886
views
Relation between Igusa tower and $p$-adic modular forms
As the title suggests, my question is devoted to understand (and maybe get some good references) the relation between the Igusa tower for a modular curves and $p$, or maybe $T$-adic modular forms. I ...
- 445
5
votes
1
answer
447
views
General existence theorem for cup products
I'm curious if it is possible to formulate cup products and prove that they exist in a general way which would subsume a lot of examples: e.g. group cohomology, sheaf cohomology for sheaves on ...
- 2,968
1
vote
0
answers
394
views
Finite generation of canonical ring in Geometric PDE
We say that a projective variety $X$ is of general type if the Kodaira dimension is equal to the dimension of $X$., i.e. $\text{kod}(X)=\dim X$.
When $K_X$ is positive then by the result of S.T.Yau ...
- 405
7
votes
1
answer
344
views
Gauss map of general K3 surface
Let $S\subset\mathbb{P}^g$ be a polarised smooth projective K3 surface of genus $g$ (and degree $d=2g-2$) over $\mathbb{C}$. Denote by $\phi: S\to G(3,g+1)$ the Gauss map, taking a point $s\in S$ to ...
- 2,956
3
votes
0
answers
133
views
Topological criterion for GIT semistability
Let $X$ be a complex algebraic variety and $G$ a complex reductive group acting on $X$. Let $L$ be a linearization of this action, i.e. a line bundle with a linear action of $G$ covering that of $X$. ...
- 31
4
votes
1
answer
448
views
Mirror symmetry for blowups of the projective plane
Let $S$ be a blowup of the projective plane $\mathbb{CP}^2$ at $n$ points. When $n\le 9$, Auroux, Katzarkov and Orlov showed that them a mirror Landau-Ginzburg model is given by a certain elliptic ...
- 41
4
votes
1
answer
486
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Intersection number of divisors on abelian surfaces and its invariance under translation by 2-Torsion points
I am reading Fulton's but I cannot find a useful result for my problem that seems to be something well known.
Let $\mathcal{J}$ be the Jacobian of a hyperelliptic curve of genus $2$. Consider $D_1,...
10
votes
1
answer
418
views
Relations for the algebra of differential operators on a smooth affine variety
Over a ground field of characteristic zero, the algebra of differential operators $\mathcal{Diff}(X)$ on a smooth affine variety $X$ is generated by the functions $O_X$ and the derivations $Der(O_X)$, ...
- 46.8k
4
votes
0
answers
504
views
The lisse-etale site and derived algebraic geometry
If one reads say Olsson's book on algebraic stacks or Laumon-Moret-Bailly. The lisse-etale topology is used to define quasi-coherent sheaves and the cotangent complex (or rather cutoff's of the ...
- 41
2
votes
0
answers
434
views
A Grothendieck style reference for probability theory
This is a very vague question.
I am looking for an "EGA" type of reference for probability theory. This means (among other things) that I'm looking for a text which develops the theory "abstractly".
- 29
5
votes
0
answers
454
views
Formal multidimensional Taylor series expansion over commutative rings
If $F:V\to W$ is a smooth at $a\in V$ function between finite-dimensional vector spaces over $\mathbb{R}$, then we have
$$
F(x) = \sum_{k=0}^N\frac{1}{k!}(D^kF)(a)[(x-a)^{\otimes k}]+\text{remainder},
...
- 6,730
9
votes
2
answers
1k
views
Galois invariant Picard group elements
Let $X$ be a smooth variety over a perfect field $k$ with $X(k) \neq \emptyset$. Then is the natural map
\begin{equation}
\mathrm{Pic}(X) \to (\mathrm{Pic}(X_{\bar{k}}))^{\mathrm{Gal}(\bar{k}/k)} \...
- 20.8k
5
votes
1
answer
397
views
Simplicial complex construction from given Betti numbers?
Is it possible given a set of Betti numbers to construct a (possibly set of) simplicial complex with the given Betti-described topology? I understand there can be an infinity of simplicial complexes ...
- 175
11
votes
2
answers
2k
views
Idea behind Grothendieck's proof that formally smooth implies flat?
From this answer I learned that Grothendieck proved the following result.
Theorem. Every formally smooth morphism between locally noetherian schemes is flat.
The book Smoothness, Regularity, and ...
- 10.3k
2
votes
1
answer
213
views
Some calculations on cones
I am reading Chapter 3 of the book "Singularities of the minimal model program", and I have some problems on calculating certain quantities.
Let $X$ be a projective variety, $L$ be an ample line ...
- 3,362
3
votes
0
answers
160
views
consequence of the definition of perverse sheaves
I am trying to learn perverse sheaves. They are complexes $M$ of sheaves with constructible cohomology (say we are working with algebraic varieties and the transcendental topology) such that the ...
- 31
3
votes
1
answer
289
views
When is the Poincare-polynomial of an element in the Grothendieck ring the actual Poincare polynomial of a representative?
There is a well-defined map
$$P:K(Var)\to \mathbb{Z}[t]$$
which sends smooth projective varieties to their Poincare polynomial. This is in fact enough to define $P$ on all elements of $K(Var)$.
I ...
- 2,768
4
votes
0
answers
186
views
Universal vectorial bi-extension as a scheme
In 'The universal vectorial Bi-extension and p-adic heights' Coleman works with the pullback of the Poincaré biextension of an abelian variety A to its universal vectorial extension and claims this is ...
- 1,437
3
votes
1
answer
167
views
Degeneration of coadjoint orbits
Let $X$ be a projective manifold and we have a degeneration of fibers such that they are biholomorphic to coadjoint orbits, i.e, $X\to \Delta$ , and fibers $X_t$ are biholomorphic to coadjoint orbits ...
- 39
7
votes
1
answer
643
views
nef vs. 1-nef vector bundles
Let $X$ be a compact, connected, Kähler manifold, of dimension $d$, with Hermitian metric $\omega$; let $E$ be a vector bundle on $X$ of rank $r\geq2$.
By [1] definition 3.1.2:
A line bundle $L$ ...
- 828
6
votes
2
answers
504
views
First Chern class of the universal bundle
Let $M$ be the moduli space of semistable vector bundles of fixed determinant $L$ and rank $r$ over a smooth curve $X$. Assume that $gcd(r,deg(L))=1$. Let $\mathcal U$ be the universal bundle over $M\...
- 1,871
8
votes
0
answers
196
views
Spin structures and bitangents
I came across an interesting remark in Atiyah's classic paper, Riemann surfaces and spin structures. He computes that there are 28 spin structures on a genus 3 surface with Arf invariant 1, and then ...
- 19.2k
2
votes
0
answers
182
views
Stein subspaces of polydiscs and balls
Let $D$ be a either an open polydisc or an open ball in $\mathbf{C}^n$.
(1) Let $\mathcal{O}$ be the $\mathbf{C}$-algebra of holomorphic functions on $\mathbf{C}^n$, resp. $D$, and let $f_1,\ldots, ...
user113452
4
votes
0
answers
339
views
Is the relative moduli space of semi-stable sheaves on families of curves fine
Let $\pi:X \to B$ be a family of smooth, projective curves. Fix coprime integers $r,d$. Denote by $\mathcal{M}(r,d)$ the relative moduli functor corresponding to rank $r$, degree $d$, semi-stable ...
- 2,126
2
votes
1
answer
199
views
Identification of cohomology sheaf in the definition of the Kodaira-Spencer morphism for abelian schemes
Let $p:A \to S$ be a projective abelian scheme, where $S$ is some smooth scheme over a base field $k$. Then we have the Kodaira-Spencer morphism
$$
\kappa : T_{S/k} \to R^1p_*T_{A/S}
$$
where $T_{S/k}$...
user85435
1
vote
1
answer
106
views
Sum of certain decomposable elements
Let $V$ be be a vector space of dimension $m$ over any field and $\ell\leq m$ be a positive integer. Let $\omega_1,\ldots,\omega_r \in\bigwedge^\ell V$ are linearly independent, completely ...
- 179
1
vote
1
answer
327
views
Automorphism group of fiber products of schemes
Let $A \mapsto S$ and $B \mapsto S$ be two schemes over the scheme $S$. Is there a connection between the automorphism group of the scheme $A \otimes_{S} B$ and the automorphism groups of $A$ and $B$ ?...
- 4,353
3
votes
0
answers
149
views
Determinant of the universal bundle
Let $M$ be the moduli space of semistable vector bundles of fixed determinant $L$ and rank $r$ over a smooth curve $X$. Assume that $gcd(r,deg(L))=1$. Let $\mathcal U$ be the universal bundle over $M\...
- 1,871
10
votes
2
answers
485
views
Limit of decomposable bundles
Let $(E_b)_{b\in B}$ be a family of vector bundles on a smooth projective variety $X$, parameterized by a smooth curve $B$. Let $\mathrm{o}\in B$. Assume that $E_b$ is decomposable (= direct sum of ...
- 37.3k
4
votes
1
answer
275
views
Vector groups in Ogus-Vologodskys Nonabelian Hodge theory in characteristic p
i hope this is the right place for this.
In https://math.berkeley.edu/~ogus/preprints/anonhodge.pdf on page 15 Ogus and Vologodsky state the following:
Let $\pi_T: \textbf{T} \rightarrow X$ be a ...
- 73
1
vote
0
answers
90
views
Partition of sets of monomials
Given a degree $d>0$ and $x_1,\dots,x_n$, consider the set $S$ of monomials
\begin{equation*}
x_1^{i_1} \cdots x_n^{i_n}
\end{equation*}
with $0\leq i_j\leq d$ for $1\leq j\leq n$ (the exponents ...
- 53
1
vote
0
answers
312
views
Cohomology of a structure sheaf of a normal affine variety
I can't find the reference for the following fact:
Let $X$ be an affine variety and let $Y$ be its smooth resolution. $H^0(X,\mathcal{O}_x)=H^0(Y,\mathcal{O}_Y)$ if and only if $X$ is normal.
- 371
13
votes
0
answers
259
views
Alexander modules and weight filtrations
$\def\Ext{\mathrm{Ext}}$I have the following situation: I have a complex algebraic variety $X$. I want to compute the weight filtration on various abelian covers $Y \to X$. As a concrete example, take ...
- 151k
3
votes
1
answer
384
views
Torsors of pushforward group schemes
I'm reading William Waterhouse's "Discriminants of etale algebras and related structures", and he makes a basic claim I'm struggling to justify.
Suppose $S/R$ is etale of rank $n$... and let $\pi$ ...
- 2,336
11
votes
0
answers
245
views
Reference for $E_{\infty}$ algebra structure on cohomology of a site
Let $A$ be a sheaf of commutative rings on a site $X$. (In my applications, $X$ comes from one of the standard Grothendieck topologies on algebraic varieties.) It should be true that $R\Gamma (X, A)$ ...
- 2,769
6
votes
2
answers
1k
views
Are irreducible subgroups Zariski-dense?
A subgroup $H$ of an algebraic group $G$ is said to be Zariski-dense if its Zariski closure is all of $G$ (or alternatively, if every polynomial which vanishes on all elements of $H$ vanishes ...
- 927
0
votes
0
answers
63
views
Generically parameterize quartics in arbitrary dimension
Suppose $x=(x_1,\dots, x_d) \in \mathbb{R}^{d}$ for an arbitrary dimension $d$. Let $p(x_1,\dots, x_d)$ be a degree 4 polynomial and consider the quartic defined by $p(x_1, \dots, x_d)=0$.
Is it ...
- 337
1
vote
0
answers
244
views
Section ring $R(X,D)$ of $D$ is finitely generated if $\kappa (X,D) \leq 1$
In the remark on the bottom of page 5 of this paper, the author states that
It is well-known fact that the algebra of a Divisor $D$ with $\kappa (X,D) \leq 1$ is finitely generated over $k$. In ...
- 61
16
votes
1
answer
704
views
Where was $I_x/I_x^2$ first introduced? (DG or AG)
Cotangent space appears in both differential geometry (DG) and algebraic geometry (AG).
In DG, given a smooth manifold $M$ and $x\in M$ one has an isomorphism $I_x/I_x^2 \cong T^*_xM$, where $I_x$ is ...
- 1,605
0
votes
1
answer
358
views
Existence of a model over S-integers
Let $G$ is a connected reductive group over a number field $F$. Let $S$ be a set of places containing the archimedean places for which $G$ has a model over the ring of $S$-integers in $F$. Is it true ...
6
votes
1
answer
393
views
Frobenius automorphisms of cohomology of a variety
Suppose $X$ is a smooth variety defined over $\mathbb{Q}$. There are (at least) two automorphisms of cohomology groups of $X$ that are called "Frobenius", and I would like to understand how they are ...
- 8,941
0
votes
0
answers
206
views
Does there exist a homogeneous polynomial $F$ whose partial derivatives satisfy the following inequalities?
Given $n \geq 2$, I would like to find either a homogeneous polynomial $F \in \mathbb{Q}[x_1, \ldots, x_n]$ of degree $d > 1$ with the following properties:
$W = \{ \mathbf{x} \in \mathbb{R}^n : F(...
- 3,595
7
votes
1
answer
472
views
Motivic $\mathbf{Z}(1)$
I know that the Bloch higher Chow complex, $\mathbf{Z}(i)_{\mathcal{M}}$, on a smooth scheme over a field $k$, reads, in degree $1$:
$$\mathbf{Z}(1)_{\mathcal{M}}\simeq\mathbf{G}_m[-1].$$
How to see ...
user113393
3
votes
0
answers
223
views
Serre duality graded singularity category
Let $R$ be a local Gorenstein ring of Krull dimension $d$ with an isolated singularity. Defined $D_{sing}(R)$ as the Verdier quotient $D^b(R)/Perf(R)$). Then, a famous result of Auslander says that ...
- 7,200
5
votes
0
answers
279
views
When is the strict transform of very ample divisor ample?
Let $X$ be a projective variety and $Y \subset X$ is a very ample divisor on $X$. Let $Z \subset X$ be a regular subvariety (but $X$ need not be regular along $Z$) of codimension $2$ and $\pi:\tilde{X}...
- 2,126
0
votes
0
answers
271
views
Union of varieties
Let $Q_1, \ldots, Q_k$ and $P_1,\ldots, P_m$ be irredicable homogenous polynomials in $\mathbb{C}[x_0,\ldots, x_n]$ such that $V(Q_1, \ldots, Q_k) \subseteq \cup_i V(P_i)$. Here $V$ is projective ...
- 2,683