Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology.

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4
votes
1answer
163 views

Does derived equivalence of the fibres imply derived equivalence of the total spaces?

Let $f:X\to B$ and $g:Y\to B$ be smooth morphisms of complex projective varieties. Assume that for every closed point $b\in B$, the fibres $X_b=X\times \kappa(b)$ and $Y_b$ are derived equivalent. ...
2
votes
0answers
53 views

Determine the representation given by space of sections of symmetric products of cotangent bundle of projective plane

In a recent project, it was interesting for me to determine the $PGL(3)$ representation given by $H^0(S^2(\Omega(1)) \otimes \mathcal O(4))$ on $\mathbb P^2$. I did this by using the Euler sequence, ...
2
votes
0answers
205 views

Why is the normalization of a general fiber the general fiber of the normalization?

Suppose $X \rightarrow Y$ is a map of reduced connected projective schemes of finite type over an algebraically closed field of characteristic 0, where $Y$ is a smooth connected curve. Let $Z ...
1
vote
1answer
103 views

Derived equivalence of families of dual abelian varieties

Let $B$ be a smooth projective complex variety and $\pi:X\to B$ a smooth projective map whose fibres $X_b$ are abelian varieties. Let $\psi:Y\to B$ be the naturally associated bundle such that the ...
4
votes
0answers
145 views

Does the cohomology comparison part of GAGA hold over the reals?

If $k=\mathbb{C}$, $X$ is a projective (or even proper) scheme over $k$ and $F$ a coherent sheaf on $X$, then there is an isomorphism $$ H^p(X,F)\rightarrow H^p(X^{an}, F^{an}) $$ (and there is also ...
0
votes
0answers
52 views

Non-local differentially smooth algebra

Let $A$ be a noetherian commutative algebra over a perfect field $k$. The algebra $A$ is said to be differentially smooth over $k$ if (1) $\Omega^1_{A/k}$ is a projective $A$-module, and (2) the ...
5
votes
2answers
201 views

Poisson structures on non-smooth manifolds with singularities

It's very known how we can describe a Poisson structure on a manifold $M$, where $M$ is a smooth manifold, but what about a Non-smooth manifold with singularities? In section $(2)$ of the paper The ...
3
votes
3answers
212 views

Is $Pic^0(X)$ of a curve of genus $\geq 1$ over a non-algebraically closed field still non-finitely generated?

Qing Liu's "Algebraic Geometry and Arithmetic Curves" page 299 COrollary 7.4.41 gives the following result. Let $X$ be a smooth, connected, projective curve over an algebraically closed field $k$, of ...
27
votes
0answers
661 views

Grothendieck's “List of classes of structures”

In Lawvere's article Comments on the Development of Topos Theory, the author writes: Similarly, Grothendieck and others unerringly recognized which kinds of mathematical structures are 'preserved ...
3
votes
1answer
354 views

Confusion about a result on Shimura and Teichmüller curves

It is shown by M. Moeller (M. Moeller, Shimura- and Teichmüller curves) that there are only 2 Shimura and Teichmüller curves in the moduli space of curves $M_g$, namely, the ones given by ...
6
votes
0answers
137 views

Bogomolov-Beauville-Fujiki form, algebraically

Let $M$ be a compact hyperkahler manifold, i.e. a manifold with three complex structures $I,J,K$ defining an action of quaternions on the tangent bundle and a metric which is Kahler with respect to ...
3
votes
1answer
161 views

Maximal ideals of polynomial ring containing a fixed element

We know that for a field $k $ and $f\in k [x]$, the only maximal ideals of $k [x]$ containing $f $ are the ideals generated by prime factors of $f $. Now, I want to know that if $R $ is an arbitrary ...
3
votes
1answer
194 views

Existence of pencils on some special curves of genus 10

Everything over $\Bbb{C}$. Say we have a smooth curve $C$ of genus $10$ which is a double cover of a smooth plane cubic curve. Therefore $C$ admits a 1-dimensional family of pencils of degree 4 ...
14
votes
1answer
588 views

Is there a Serre intersection formula in analytic geometry?

There is the famous Serre intersection formula in algebraic geometry using the Tor functor (see for example here). I would like to know if there is such a formula in analytic (i.e. complex) geometry. ...
1
vote
1answer
91 views

Conormal bundle of the strict transform

I'm trying to understand the proof of Theorem II.1.7 in Kollar's Rational curves on algebraic varieties. In particular, there is a claim in there that I can't make sense of. The setting is the ...
6
votes
2answers
821 views

Does module Hom commute with tensor product in the second variable?

Let $A$ be a commutative ring, and $L, M, N$ be $A$-modules. Then is it true that $$\text{Hom}_A (L, M)\otimes_A N \cong \text{Hom}_A (L, M\otimes_A N)$$ as $A$-modules? (Note that there is a ...
7
votes
2answers
205 views

The Picard number of the Kummer surface of an abelian surface

Let $A$ be an abelian surface and $\text{Km}(A)$ be the Kummer surface of $A$. If I remember correctly, the Picard number $\rho(\text{Km}(A))$ is equal to $16+\rho(A)$. Does anyone know any ...
4
votes
1answer
95 views

$Ex(f)$ has codimension at least 2

The following is a part of proof of lemma 6.2 in the book. $f:X \to Y$ a projective birational morphism of normal varieties $D$: Weil divisor on $Y$, $E$: exceptional divisor of $f$ ...
2
votes
0answers
96 views

Elliptic fibration arising from a higher genus linear system

Let $H$ be a very ample linear system on a smooth compact complex surface $X$ whose Kodaira dimension is $\geq 0$. A general element of $H$ is smooth and has genus $\geq 2$. Let $L\subset H$ be a ...
1
vote
0answers
87 views

Any natural examples of infinite dimensional Cohomological Field Theories?

Cohomological Field Theories as defined by Kontsevich and Manin are a class of linear maps from a vector space $V$ to the cohomology of the Deligne-Mumford moduli space of curves ...
1
vote
0answers
71 views

Irreducible real curves on ${\mathbb C}P^1$ invariant under the finite group action

Let $G$ be a finite subgroup of a Möbius group with a standart action on a real algebraic variety ${\mathbb C}{\mathbf P^1}.$ How one can describe $G$-invariant irreducible real algebraic curves? ...
2
votes
0answers
82 views

How the existence of holomorphic sections depends on the choice of complex structure

In this Mathoverflow question it is asked how many invariant complex structures exist on the full flag manifold of $SU(m)$. In this question it is asked when a line bundle over a flag manifold has ...
3
votes
1answer
183 views

A Bertini-type result for hypersurfaces containing a subvariety

Let $P$ be a smooth projective variety of dimension $4$ and let $Z$ be an irreducible subvariety of dimension $2$ ($Z$ is not necessarily smooth, but you can assume it). Is there a smooth, ...
5
votes
0answers
105 views

Nonclassical polynomials, circles, and groups

Tao and Ziegler have introduced a generalization of polynomials over a prime field called nonclassical polynomials, useful for studying the Gowers norm. A nonclassical polynomial of degree $d$ is a ...
3
votes
0answers
271 views

A relation between a ring with its polynomial ring

Let $\{f_i(x)\}_{i\in I}$ be a subset of $R[x]$ where $R[x]$ is the polynomial ring of $R$(a commutative ring with identity). If the ring $R/\langle f_i^2(n)-f_i(n)\rangle_{i\in I, n\in A}$, for every ...
2
votes
0answers
102 views

Lifting group actions on a gerbe

Suppose $C$ is a smooth projective curve, over a number field. Let $G$ be its group of automorphisms. Suppose $\alpha$ is a class in $H^2(C,\mu_2)$ left fixed by $G$. Let $\mathcal{C}\rightarrow C$ be ...
6
votes
1answer
327 views

Is the sheaf of smooth functions flat?

Let $X$ be a smooth algebraic variety over $\mathbb{C}$. Is the sheaf of smooth functions on $X$ flat as an $\mathcal{O}_X$ module?
7
votes
0answers
127 views

Topology of family of complex varieties

It seems to be an oft-cited fact (which comes up, for instance, in describing vanishing/nearby cycles) that: For a proper flat map $f \colon X \rightarrow \Delta$, where $X$ is a complex ...
3
votes
1answer
106 views

Torsion point on jacobian of ramified cover

Suppose $C$ is a hyperelliptic curve. Then the set of two-torsion points on the jacobians is generated by the set of difference of Weierstrass points. Suppose $C'$ is another hyperelliptic curve. Is ...
12
votes
3answers
590 views

Rational solutions of the Fermat equation $X^n+Y^n+Z^n=1$

As a generalisation to the equation of Fermat, one can ask for rational solutions of $X^n+Y^n+Z^n=1$ (or almost equivalently integer solutions of $X^n+Y^n+Z^n=T^n$). Contrary to the case of Fermat, ...
3
votes
1answer
172 views

Rationality of higher dimensional du Val singularities

I am interested in the isolated singularity defined over $\mathbb{C}$ by $$ x_1^2+\cdots + x_n^2+x_{n+1}^k=0, $$ where $n>2$ and $k>2$. I would like to know whether this singularity is ...
13
votes
1answer
522 views

Motivation and potential applications of spectral algebraic geometry

Nowadays there is a lot of talk about derived algebraic geometry, but not so much about the related subject of spectral algebraic geometry. Now I'm curious what future is there for spectral algebraic ...
21
votes
1answer
844 views

A geometric theory of Blueprints? (Algebras over the field with one element)

In my attempt to tackle the various approaches of defining algebraic geometry over $\mathbb F_1$, I was just reading through Lorscheid's paper The geometry of blueprints. I certainly like the idea a ...
16
votes
0answers
290 views

What should motives for $L(E,n)$ look like?

Goncharov and Manin showed in this paper that the zeta values $\zeta(n)$ can be realized as periods of framed mixed Tate motives constructed from moduli spaces $\overline{\mathcal{M}}_{0,n+3}$ of ...
2
votes
1answer
256 views

Dimension of a commutative ring

For a commutative ring $R $ with identity, if $Nil (R)\not= J (R)$, can we deduce than $dim (R/J (R))<dim (R)$? ($R $ has finite $Krull$ dimension (=$dim$), $Nil (R)$=the set of all nilpotent ...
1
vote
0answers
150 views

Non-vanishing of a higher direct image

Let $f:X \to Y$ be a small birational morphism between threefolds. Assume $X$ has terminal singularities and the relative Picard of $f$ is $1$. Suppose the exceptional locus of $f$ is a curve $C$, and ...
53
votes
1answer
1k views

$R$ is isomorphic to $R[X,Y]$, but not to $R[X]$

Is there a commutative ring $R$ with $R \cong R[X,Y]$ and $R \not\cong R[X]$? This is a ring-theoretic analog of my previous question about abelian groups: In fact, in any algebraic category we may ...
5
votes
2answers
95 views

(Re)construction of a polygon from all inter-vertex distances

For a $n$-polygon in $\mathbb{R}^3$ the set of distances between all pairs of vertices is given. (How) is it possible to reconstruct the geometric structure of the polygone? Symbolically: For a set ...
1
vote
1answer
254 views

maximal ideals of polynomial ring

For a maximal ideal $ n $ of polynomial ring $ R [x] $ over a commutative ring $R$ with identity, are there conditions under which $m [x]\subset n$, for some maximal ideal $m$ of $R$?($m [x] $ is ...
1
vote
1answer
82 views

What is the Fano index for Hermitian symmetric spaces of compact type?

As we know Hermitian Symmetric spaces of compact type are all Fano picard number one, we can talk about his Fano index. Suppose $X$ is one of those Hermitian symmetric spaces, $L$ is the generator of ...
2
votes
2answers
213 views

$\{$Affine schemes over $S$$\}$ $\cong$ $\{$$\mathcal{O}_S$ - algebras$\}$?

I've asked something very similar before on MSE but unfortunatly it hasn't recieved a lot of attention. I decided to ask again here. Let $S$ be a fixed scheme. Is the following true? "Theorem": ...
1
vote
0answers
66 views

A problem related to parametrizing $\operatorname{rank}\le r$ matrices and Segre embedding

Given a field $k$. We denote $A_{mn}=k[\{X_{ij}\}_{1\le i\le m,1\le j\le n}]$ a polynomial ring of $mn$ variables. Given $m,n,r>0$, we have a natural homomorphism $\phi\colon A_{mn}\to ...
4
votes
0answers
151 views

Derived equivalent varieties with differing integral Mukai-Hodge structures?

For a smooth projective complex variety $X$ of dimension $n$, let $H^i(X)$ denote its integral Hodge structure of weight $i$. Define $\tilde{ H^0}(X) = \bigoplus H^{2i}(X)\otimes \Bbb Z(i)$ and ...
8
votes
0answers
147 views

List of known Fourier Mukai partners?

I'm familiar with some examples of pairs of derived equivalent varieties, for example an abelian variety and its dual, a K3 surface and certain moduli schemes on it, or the Pfaffian-Grassmannian ...
3
votes
2answers
130 views

Are curves with maximal Clifford index Brill-Noether general?

By the Brill-Noether Theorem, a general curve $C$ of genus $g\geq2$ has maximal Clifford index $\lfloor \frac{g-1}{2}\rfloor$. Hence a very naive question is: (Q1) Is a curve with maximal Clifford ...
10
votes
1answer
433 views

teaching higher algebra

Has anyone ever (successfully or unsuccessfully) taught a course in higher algebra (in the $\infty$-categorical sense)? I'm asking out of curiosity (and also hoping for more resources). The ...
4
votes
0answers
246 views

Moduli of coherent sheaves on abelian varieties

Let $\mathcal{A}_g$ be the moduli space (stack?) of $g$-dimensional principally polarized abelian varieties. We have the universal family of abelian varieties $\chi_g\rightarrow \mathcal{A}_g$, where ...
3
votes
0answers
97 views

Construction of algebraic curves using line bundles on graphs

In this paper http://arxiv.org/abs/0707.1309 Matthew Baker and Serguei Norine, construct a analogue of the Riemman Roch formula for Lineal Systems defined on graphs. In the paper ...
5
votes
1answer
208 views

Generalized Euler characteristics of non-motivic origin

By a generalized Euler characteristic $\chi$, I mean an isomorphism invariant $\chi(V)$ inside some abelian group $A$, defined for every varietiy $V$ over a field $k$, with the property that, for all ...
5
votes
0answers
149 views

Homogeneous spaces of affine algebraic groups

Let $G$ be a reductive algebraic group over an algebraically closed field $K$ of characteristic zero (I am particularly interested in the case $G=GL_n(K))$. Let $H$ be a closed subgroup of $G$. It is ...