Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology.
-2
votes
0answers
31 views
The symmetric powers of the $O_{X}-$modules
Let $C$ be a smooth projective curve of genus $g\geq 1$. The symmetric powers of the curve $C$ are
$$
Sym^{r}(C)=(C\times\dots\times C)/S_{r},
$$
where $S_r$ denotes the symmetric group of degree $r$.
...
0
votes
0answers
21 views
moduli space of equivariant holomorphic embeddings into the quintic
I'd like to understand better the following problem, whether it is mathematically well-posed, trivial, etc.
Fix a non-negative integer $g$ and consider
the space
...
1
vote
2answers
82 views
Automorphisms of a smooth quadric surface $Q\subset\mathbb{P}^{3}$
Let $Q\cong\mathbb{P^{1}_{1}}\times\mathbb{P^{1}_{2}}\subset\mathbb{P}^{3}$ be a smooth quadric surface. We have the following two actions on $Q$:
$$S_2\times Q\rightarrow Q,\; ...
8
votes
1answer
289 views
Why the name “variety” and the notation “V” for zeroes of polynomials?
The following questions came to my mind while preparing the notes for the first class of (my first) course on algebraic geometry.
Question 1: Is there any motivation for choosing the term "variety" ...
5
votes
0answers
34 views
Bounding volume of cell in complement of zero set
I am given an integer polynomial $f \in \mathbb{Z}[X_1, \ldots, X_n]$ of bounded degree and bounded coefficient size. The polynomial's zero set partitions $\mathbb{R}^n$ into cells. What I am looking ...
-2
votes
0answers
95 views
Can infintly generated algebras be smooth?
Is it possible for an infinte dimensional algebra to be smooth?
That is can an infintly generated algebra have finite global dimension?
For example is the polynomial ring $R[x_n]_{n \in \mathbb{N}}$ ...
-1
votes
1answer
89 views
is the fixed point locus integral and reduced?
Let $X$ be a scheme over a field $k$ of characteristic zero and let $G$ be finite group acting on $X$. Then one can define the scheme $X^G$ of fixed points of $X$. It is a closed smooth subscheme of ...
0
votes
0answers
122 views
tensor of motives
Let $X$ and $Y$ be two smooth projective varieties over $k$. Consider the Chow motives $M(X\times\mathbb{P}^n)\simeq M(X)\otimes M(\mathbb{P}^n)$ and $M(Y\times\mathbb{P}^n)\simeq M(Y)\otimes ...
4
votes
1answer
126 views
Chow ring of two varieties
Suppose we are given two smooth projective varieties $X$ and $Y$. Maybe this is elementary but what is the Chow ring $CH(X\times Y)$ in terms of $CH(X)$ and $CH(Y)$?
1
vote
1answer
105 views
Direct image of an ideal sheaf along a blow-up
Suppose that $I\subseteq\mathbb{C}[x_0,\ldots,x_n]$ is a saturated homogeneous ideal. Let $\mathcal{I}\subseteq\mathcal{O}_{\mathbb{P}^n}$ denote the corresponding coherent ideal sheaf, and then let ...
1
vote
1answer
81 views
endomorphisms of the Jacobian of a curve
Let $C$ be a smooth, projective curve over the complex numbers and let $J(C)$ be its Jacobian. The Torelli theorem relates the automorphisms of $C$ to the automorphisms of $J(C)$. Precisely, ...
-4
votes
0answers
47 views
$U(1)$ quotient of sphere $S^{2N-1}$ is projective space [on hold]
I am physical background,and don't have much algebra geometry knowledge.
There is a statement in the textbook that
'$\mathbb{C}P^{N-1}$ is the $U(1)$ quotient of sphere $S^{2N-1}$'.
Could someone ...
10
votes
2answers
308 views
Microlocalizing Hochschild homology
A recent paper of Ben-Zvi and Nadler gives a general formalism for understanding "dimensions" in sheaf theories. Without getting too far into details, amongst other things, this formalism allows us ...
-1
votes
0answers
59 views
is the structure sheaf of a variety with quotient singularities reflexive? [on hold]
The question is kind of contained in the title. Let me say a few more words. I am interested in varieties $X$, say over the complex numbers, with only quotient singularities. Instead of taking the de ...
1
vote
1answer
136 views
blow up and derived category
Consider the blowup $X$ of $\mathbb{P}^2$ at a single point $p$. Then, Orlov showed that there is a semiorthogonal decomposition $D^b(X)=\langle e,O_X,O_X(1),O_X(2)\rangle$, where $O_X(i)$ is the ...
9
votes
0answers
120 views
real algebraic geometry software?
Does anyone have suggestions/experience for any software packages to study real algebraic varieties (for example, counting connected components of hypersurfaces, figuring out the topological type of ...
5
votes
0answers
107 views
Is there a holomorphic Morse-Bott lemma?
It asks for a generalization of the question in the post
Normal form for a holomorphic Morse function
Suppose $f$ is a holomorphic function on a complex manifold $M$ which has Bott type critical ...
1
vote
1answer
136 views
Is a semisimple conjugacy class closed?
Let $G$ be any algebraic subgroup of $\mathrm{GL}_n$ over an algebraically closed field of any characteristic.
If $s$ is a semisimple element of $G$, can the $G$-conjugacy class of $s$ fail to be ...
0
votes
0answers
49 views
Short exact sequence of sheaves and intersection of curves [migrated]
Let $C_1$ and $C_2$ be two (smooth rational) curves, $D=C_1+C_2$, $C_1\cap C_2=p$ (a single point). Then how can I show that there is a short exact sequence of sheaves
$
0\rightarrow ...
7
votes
2answers
215 views
Necessary condition on Calabi-Yau manfiold to be a hypersurface in a Fano manifold
Let $X$ be a smooth projective Calabi-Yau threefold. Are there any known obstructions to it
being a member of a base-point-free linear system in a nef-Fano fourfold?
What, in anything, is known ...
3
votes
0answers
104 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$ ...
4
votes
0answers
114 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 < ...
5
votes
1answer
426 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
129 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 ...
5
votes
2answers
196 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 ...
1
vote
1answer
109 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
88 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
62 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}$. ...
5
votes
2answers
250 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
129 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
141 views
commutative diagram with Yoneda pairing, Weil pairing and edge morphism
Why does the following diagram commute?$\require{AMScd}$
\begin{CD}
H^0(X,\mathscr{A}) \times \mathrm{Ext}^2_X(\mathscr{A},\mu_{\ell^n}) @>>> H^2(X,\mu_{\ell^n}) \\ @VVV @| \\
...
0
votes
2answers
91 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 ...
3
votes
1answer
115 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
90 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
88 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
72 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
204 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
324 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
104 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
64 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
93 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. ...
2
votes
2answers
163 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
37 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 ...
7
votes
1answer
162 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 ...
13
votes
2answers
424 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
86 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").
7
votes
1answer
247 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
53 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
67 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 ...
