Tagged Questions
Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology.
0
votes
0answers
28 views
Does the following first order approximation of the Kapranov-Vasserot infinitesimal loops still do any job?
Let $X$ be a scheme over, say, a field $k$. Let us denote $\mathrm{Spec}(k[\varepsilon])$ by $T$ and its (unique) $k$-point by $0\in T$. Call the first order infinitesimal cone $C_{T,0}(X)$ over $X$ ...
1
vote
0answers
57 views
Kodaira vanishing for Du Bois singularities
Let $X$ be a projective variety with Du Bois singularities, which is additionally assumed to be Cohen-Macaulay. Then $H^i(X, \mathscr L^{-1}) = 0$ for any ample line bundle $\mathscr L$ and $i < ...
4
votes
1answer
361 views
Main conjecture for elliptic curves
Suppose that $E$ is an elliptic curve defined over $\mathbb{Q}$, and that $p$ is a prime where $E$ has good ordinary reduction. Then one can define nonnegative integers $ \lambda_{E}^{alg} $, $ ...
2
votes
1answer
109 views
Counting curves of degree 4 in $\mathbb{P}^{3}$
Let $p_1,...,p_8\in\mathbb{P}^{3}$ be points in linear general position. Then there exists a unique elliptic curve $C$ of degree $4$ passing through $p_1,...,p_8$. I am interested in what happens for ...
4
votes
2answers
133 views
Formal completion of the normal bundle
Let me for simplicity start with affine case. If $X=\operatorname{Spec}(A)$ is an affine variety $Z \subset X$ is a closed affine subvariety $Z=\operatorname{Spec}(A/I)$. What conditions are ...
-4
votes
0answers
157 views
Do you know their definitions? [on hold]
contragredient representation
extremal weight
fine K-type
fine representation
generalized infinitesimal character
good roots
Harish-Chandra map
Harish-Chandra module
Hochschild-Serre spectral ...
1
vote
1answer
100 views
Deformation space form the point of view of intersection theory
I'm interested in deformations of subvarieties of a toric variety $X$.
Suppose we know a subvariety $V$ in the Chow group of $X$, for example, $V$ is a linear combination of powers of hypersurfaces. ...
0
votes
0answers
75 views
Inducing a comodule structure on Hom
If $C$ is an $R$-coalgebra and $M$ is an $R$-module... then is it possible to endow
$Hom_{_RMod}(C,M)$ or $Hom_{_RMod}(M,C)$ with the structure of a $C$-comodule?
0
votes
0answers
59 views
cohomology of a space from a map to affine plane
Suppose $X$ is an affine variety,complete intersection in $\mathbb{C}^{2n}$, but with a high dimension of singularities. I also have a surjective finite algebraic map $f:X\rightarrow \mathbb{C}^{d}$. ...
4
votes
2answers
237 views
Making Hironaka's theorem explicit for hypersurfaces
Given a smooth hypersurface $H$ in $\mathbb{C}^n$, a theorem of Hironaka promises that one can find a strict normal crossings compactification $\bar{H}$ inside of a projective variety $X$. For me, ...
0
votes
2answers
115 views
Injective resolution for right derived functor
This question is base on my previous question, and I repeat it here:
Suppose $X$ is a projective variety and $D^{+}(X)$ is the derived cateogry of bounded below complexes of sheaf of ...
2
votes
0answers
68 views
commutative diagram with Yoneda pairing, Weil pairing and edge morphism
$\require{AMScd}$
Why does the following diagram commute?
\begin{CD}
H^0(X,\mathscr{A}) @VVV \times \mathrm{Ext}^2_X(\mathscr{A},\mu_{\ell^n}) @>>> H^2(X,\mu_{\ell^n}) @| \\
...
0
votes
2answers
84 views
Poincaré bundle and Weil pairing for Abelian schemes
In which situations is there a Poincaré bundle for Abelian schemes? In [Mumford, Abelian varieties] only the case of Abelian varieties is treated.
The same question for the Weil pairing ...
2
votes
1answer
104 views
$\mathbb{P}^1$-fibrations over $\mathbb{P}^2$ that are not rational
We know that if $X$ is a smooth complex projective variety and we assume that there is a dominant morphism $f : X \to Y$ with $Y$ and the general fibers of $f$ rationally connected. Then $X$ itself is ...
2
votes
1answer
80 views
Iterated extensions of quotients of vector bundles
Let $X$ be a noetherian scheme. Consider the smallest class $\mathcal{S}$ of coherent sheaves on $X$ which has the following closure properties:
Every locally free sheaf is in $\mathcal{S}$.
If $A,B ...
3
votes
1answer
82 views
Normalization of a curve and push forward of vector bundles
Let $C$ be a projective curve (over an algebraically closed field, not necessarily of characteristic zero) which is smooth except for exact one node. Let $\pi:\tilde{C} \to C$ be its normalization. ...
1
vote
0answers
68 views
Positivstellensatz for non-polynomial term
Can we use Positivstellensatz (P-satz) below for a non-polynomial term?
P-satz:
Let $R$ be real closed field. Let $f,g,h$ be finite families of polynomials in $R[X_{1} ,...,X_{n}]$. Denote by P the ...
2
votes
1answer
194 views
why do automorphisms preserve ample divisors?
Let $X \hookrightarrow \mathbb{P}$ be a smooth hypersurface inside some projective space $\mathbb{P}$ and let $H$ be a smooth hyperplane section of $X$. Now let $\varphi$ be an automorphism of $X$.
...
4
votes
2answers
317 views
Why is the derived tensor product only defined for bounded above derived categories?
In "Residues and Duality" by Hartshorne, the derived tensor $\otimes$ only defined for the bounded above categories (see Chapter I, section 4), that is one has
$$\otimes: D^{-}(X) \times D^{-}(X) \to ...
3
votes
1answer
98 views
geometric interpretation and differences of Gorenstein rings, Complete intersections and regular rings
R is a local Noeth.
What is geometric interpretation and of
1- Gorenstein rings
2- Complete intersections
3- regular rings?
how can I realize differences by geometric interpretation
1
vote
0answers
63 views
Algebraic set given by sequence of polynomials
When working on some problem, I have end up with a following situation. Suppose
$P(z)=z^d+a_{d-1}z^{d-1}+\ldots+a_1z+a_0$ is a complex polynomial $d\geq2$ and that $\gamma$ and $\delta$ are non-zero ...
3
votes
1answer
91 views
on a family of CM Hodge structures
I am having a look at the paper "Hodge structures of type $(n, 0, \ldots, 0, n)$" by Totaro. At the very beginning he says that a Hodge structure of type $(1, 0, 1)$ has always complex multiplication. ...
1
vote
2answers
154 views
Some questions on vanishing of Ext sheaves
Let $X$ be a complex manifold of dimension $3$ and $\mathcal{E}$ be a coherent sheaf such that $\dim supp(\mathcal{E})=1$. In this situation, I would like to know why we have the Ext sheaf ...
1
vote
0answers
35 views
Any one know the results of tensor decomposition by hypergraph partitioning?
Tucker Decomposition and CANDECOMP/PARAFAC (CP) Decomposition are two widely used tensor decomposition methods.
However, when we model the hypergraph into tensor, what's the connection between the ...
6
votes
1answer
154 views
Finite morphisms to projective space
Let $X$ be a projective variety of dimension n. Then there exists a finite surjective morphism $X \to \mathbf P^n$. Let $d$ be the minimal degree of such a finite surjective morphism.
Let $d^\prime ...
12
votes
2answers
408 views
Why is “naive” definition of non-commutative spectrum bad?
It is well-known that the category of affine schemes is equivalent to the opposit category of commutative unital rings. So naively, one would think that the same should hold in non-commutative ...
0
votes
0answers
84 views
Consistency of the u-invariant under field extension
A algebraic field extension L/k induces of homomorphism between the Wittrings. We get
$\phi: W(k) -> W(L)$. If every anisotropic isometry class of $W(k)$ stays anisotropic, the kernel of $\phi$ ...
-1
votes
0answers
42 views
emb $\dim (R) = \text{depth}(R) + 1$ then $R$ is a Gorenstein [migrated]
$(R,m)$ is a Noetherian local ring such that emb $\dim(R) = \text{depth}(R) + 1$. Show that $R$ is a Gorenstein ring.(Here emb $\dim R$ means "number of minimal system of generators of m").
6
votes
1answer
194 views
Log forms and Tate classes
Let $X$ be a smooth finite type variety over $\mathbb{C}$. Suppose that $\theta$ is a closed algebraic $1$-form whose cohomology class is weight $2$.
Can we always express $\theta$ as
$$\theta = ...
1
vote
0answers
50 views
How to Calculate Minimal Log Discrepancy on a Toric Variety?
Let $\sigma$ be a $3$ dimensional strongly convex rational polyhedral cone on $\mathbb{R}^3$ and $X_\sigma$, the corresponding affine toric $3$-fol. Also, assume that $\Delta$ is an anti-boundary ...
1
vote
0answers
66 views
Can discrepancy change after pseudo isomorphisms?
Let $X$ and $Y$ be two projective irreductive algebraic varieties of dimension $3$ and let $f:X\dashrightarrow Y$ be a pseudo isomorphism, i.e. a birational map which restricts to an isomorphism ...
1
vote
0answers
69 views
Degeneracy divisor of the “trace” morphism
Let $f\colon X\to Y$ be a finite morphism of smooth curves over an alg. closed field of characteristic zero. I recently asked how methods reminiscent of basic algebraic number theory can be used to ...
5
votes
2answers
254 views
The trace of a wedge product of matrices
I'm trying understand a computation on page 371 of Besse's book on Einstein Manifolds.
I already know the curvature operator $R:\bigwedge^2\to\bigwedge^2$ may be written in block diagonal form ...
4
votes
0answers
78 views
local systems with cyclic monodromy
In their book "Lectures on vanishing theorems", Esnault and Viehweg used finite cyclic covering of varieties constructed as follows:
Let $X$ be a smooth projective variety over some field $k$ of ...
5
votes
1answer
86 views
Properties of singularities that are preserved by categorical quotients
Let $G$ be a reductive group acting on an affine singular
variety $X$, and let $X/G$ be the categorical quotient. I know that if
$X$ has rational singularities, then so does $X/G$ ...
5
votes
1answer
157 views
Positivity question on K3 surfaces
Let $X$ be a smooth projective complex K3 surface and $L, D$ two effective divisors, $L^2\geq0$ and $D^2\geq0$.
(Q1). do we have $L\cdot D\geq0$ ?
If either one has positive self-intersection, the ...
8
votes
2answers
464 views
Vector bundles on vector bundles
Are all vector bundles on a given vector bundle the pull back of a vector bundle on the base?
In more detail: let $X$ be a space and $p:E\rightarrow X$ a vector bundle over $X$. Let $\iota: X ...
1
vote
0answers
168 views
Blowdown and contraction
I am sorry, my question is very naive.
2nd Edit: Let us suppose that $V$ is a smooth complex projective variety, and $Y\subset V$ is a smooth divisor and has an ample conormal line bundle.
We would ...
4
votes
1answer
212 views
rationality question while dealing with an isogeny
I don't think that the following is known, but before going to other things, I would like to know what can be said about it. Thanks in advance for any relevant comment !
So here is the situation. Let ...
1
vote
0answers
149 views
An optimization problem on the sphere
Let $S$ be a sphere centered at origin in $\Bbb R^{2n}$ of radius $\sqrt{2n}$. Let $D$ be a diagonal matrix. Let $U$ be an orthogonal matrix. Let $r\in\Bbb Z_+$ be a fixed integer.
Let vector ...
2
votes
1answer
151 views
Fiber products and torsors
Suppose $G$ is a finite linear group, and I have a $G$-torsor $Z \to X$. Suppose also I have a morphism $f : Y \to X$ with some properties $P$. What should these properties $P$ be in order to make the ...
5
votes
1answer
254 views
Root space decomposition
What is the root system for the special unitary lie algebra $\mathfrak{su}(p, q)$. Remind that these are matrices of the form
$\left(
\begin{array}{cc}
X & Y \\
\overline{Y}^t & Z ...
8
votes
0answers
205 views
Topological type of Brieskorn manifolds
Let us consider the complex hypersurface and suppose that $n\geq 3$:
$$F(d,n)=\{(z_0,\ldots,z_n)\in \mathbb{C}^{n+1}:z_0^d+z_1^d+\ldots+z_n^d=0\}$$
and the link $V(d,n)=F(d,n)\cap S^{2n+1}_{\epsilon}$ ...
13
votes
2answers
459 views
$p$-adic periods
For a variety $X$ defined over $\mathbb{Q}$, there's a (functorial) comparison isomorphism
$$
H^i_{dR}(X)\otimes\mathbb{C}\to H^i_B(X,\mathbb{Q})\otimes\mathbb{C}.
$$
If we pick $\mathbb{Q}$-bases for ...
3
votes
2answers
102 views
Irreducible divisor in a basepoint free linear system
Let $X$ be a projective, normal variety over complex field with canonical singularities. Suppose $|D|$ is a basepoint free linear system, then is it true that the generic elements in $|D|$ are ...
3
votes
0answers
114 views
Is the formula for plethysm $S^n(S^3)$ known explicitely?
Is the formula for plethysm (in this the decomposition into irreducible GL representations of the composition of symmetric powers) $S^n(S^3)$ known explicitely? I know $S^n(S^2)$ e.g. in Macdonald's ...
4
votes
1answer
123 views
Cocycle condition for equivariant sheaves
Let $G$ be an affine group that acts on a variety $X$. Equivariant sheaves on $X$ could be defined in the following way. Consider the simplicial space $X_\bullet$
: $X_n := G^n \times X$, $s_0:X_0 ...
5
votes
1answer
173 views
Does $h^1(D)=0$ imply numerical connectedness on K3 surfaces?
Let $X$ be a complex K3 surface and $D$ an effective divisor on $X$.
We shall say: $D$ is connected if its support is connected. $D$ is numerically connected if for any non-trivial effective ...
1
vote
0answers
76 views
Functoriality of Br(X)
Suppose $X$ is a variety over a field $k$, $k \subset K$ is a field extension, and $X_K = X \times_{Spec(k)} Spec(K)$. We have a map $X \to X_K$. By functoriality of the Brauer group, we have a map ...
7
votes
2answers
297 views
Forms of algebraic varieties
Let $X$ be an algebraic variety (say, projective, irreducible and smooth), defined over a field $K$, and let $L$ be a Galois extension. I am interested in algebraic varieties $Y$, defined over $K$, ...
