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

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90 views

### Gysin exact sequence for a singular subvariety

Let $k$ be an algebraically closed field (I'm interested in a characteristic $p>0$ specific example) and let $X$ be a (smooth if needed) algebraic variety.
Let $Y \subset X$ be a (possibly) ...

**4**

votes

**2**answers

353 views

### A systematic canonical construction of the Hodge star operator

I'm struggling to make sense of the Hodge star as a global canonical object. Here are my struggles so far and some questions:
Let $M$ be a finitely generated projective $R$-module (hence locally free ...

**7**

votes

**1**answer

215 views

### Base schemes and Bayesian priors

One of Grothendieck's dicta about algebraic geometry is to consider "the relative situation", where one doesn't consider the category of schemes but of schemes over a fixed base scheme.
In Bayesian ...

**4**

votes

**0**answers

106 views

### Extend a vector bundle on a flat family

Let $f: X\to T$ be a flat family, and $\mathcal{F}_t$ is a vector bundle on $X_t$ for some $t\in T$. Can this $\mathcal{F}_t$ be extended to a vector bundle $\mathcal{F}$ on $f^{-1}(U)$ for some open ...

**4**

votes

**1**answer

200 views

### Automorphisms and infinitesimal deformations of a smooth complete intersection

Let $X\subset\mathbb{P}^{n+c}$ be a smooth complete intersection of dimension $n$.
Is it known when $Aut(X)$ is finite ?
Does there exist a formula for the dimension of the tangent space to the ...

**10**

votes

**1**answer

179 views

### Joyal's construction of the spectrum of a commutative ring

I am trying to understand bits and pieces of Lawvere's article Continuously Variable Sets; Algebraic Geometry = Geometric Logic. I'm not doing very well.
I know this is a lot to ask, but basically, I ...

**2**

votes

**0**answers

63 views

### Can one control the ramification of a Brauer class under birational morphisms?

Assume we are given a Brauer class $\xi\in Br(k(\mathbb{P}^n))$ ramified at some divisor $D\subset \mathbb{P}^n$, here $k=\mathbb{C}$.
If $f: \mathbb{P}^n\mathrel{-\,}\rightarrow \mathbb{P}^n$ is a ...

**1**

vote

**0**answers

80 views

### Change in the number of connected components in a variety

I have a parametric variety produced by the intersection of two multinomials $V_1=0$ and $V_2=\lambda$ where $\lambda$ is a real-valued parameter. I know that for a range of $\lambda \in ...

**4**

votes

**1**answer

177 views

### Is a Laskerian ring coherent

A commutative ring $R$ with identity is said to be coherent if every f.g. ideal of $R$ is f.p. We know that any noetherian ring is coherent. A Laskerian ring is a ring in which every ideal has a ...

**12**

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**0**answers

132 views

### Coefficients of universal Schubert polynomials

Let $e_i^j$ be the elementary symmetric polynomial in $x_1,x_2,\ldots,x_j$. Then the ordinary Schubert polynomial has an expansion of the form
...

**5**

votes

**1**answer

185 views

### Birational Invariants

Let $X$ be a smooth rational variety of dimension $n$. We have $\dim H^0(X,\Omega_X^p) = \dim H^0(\mathbb{P}^n,\Omega_{\mathbb{P}^n}^p)$ for any $p$. These are Hodge numbers. I know that we can not ...

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votes

**1**answer

791 views

### Can one expect the existence of a relevant approach for a proof of the Riemann hypothesis using Mochizuki's theory? [closed]

Next month at Oxford university, there will have the first workshop outside Asia on the Inter-Universal Teichmuller theory of Shinichi Mochizuki: ...

**2**

votes

**0**answers

35 views

### How to approximate higher-degree multivariate polynomial in space of lower-degree multivariate polynomials with some constraints?

For a polynomial $P_{1}(x)$, $x\in {\mathbb R}^n$ with a higher-degree, how to find a lower-degree polynomial $P_{2}(x)$ with determined structure or bounded degree to approximate it with the ...

**8**

votes

**1**answer

507 views

+100

### Hilbert schemes and moduli of ideal sheaves

Let $X$ be a smooth projective variety over $\mathbb{C}$. The Hilbert scheme on $X$ parametrizes quotients $\mathcal{O}_X \to E$ with fixed Hilbert polynomial. Let us fix the Hilbert polynomial to ...

**3**

votes

**1**answer

163 views

### About the decomposition of a Chow group of a variety

I would like that someone helps me to find an article on the net treating the following decomposition : $ \mathrm{CH}^k (X)_{ \mathbb{Q} } = \displaystyle\bigoplus_{ i + j = k } \mathrm{CH}^{i,j} (X) ...

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votes

**3**answers

162 views

### Can an ample class be written as a sum of two classes which are not even nef?

Let $X$ be a smooth projective surface over $\mathbb{C}$. We know that nef line bundles form a closed convex cone whose interior is the ample cone. My doubt is the other direction, is it possible to ...

**0**

votes

**1**answer

119 views

### Is the dual of the dual curve the original curve? [closed]

In algebraic geometry, is the dual of the dual curve the original curve? Why or why not?

**6**

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**0**answers

289 views

### Smoothing a piecewise smooth manifold

Let $M \subset \mathbb{R}^d$ be a piecewise smooth $2$-manifold. Let $C$ be a polyhedral complex that covers $\mathbb{R}^d$ and contains faces of dimension $[0,d]$. Since $M$ is a $2$-manifold, we can ...

**4**

votes

**1**answer

158 views

### Canonical models of Teichmüller curves

It is well-known that Shimura varieties can be defined over number fields and that moreover they possess canonical model over number fields. On the other hand, Teichmüller curves can also be defined ...

**3**

votes

**1**answer

149 views

### Question about canonical DM stacks

Let $\mathcal{X}$ be a canonical stack (edit: I forgot to say I also want $\mathcal{X}$ smooth), and $\pi : \mathcal{X}\to X$ its coarse moduli space morphism. Let also $D$ be a prime divisor (i.e. ...

**4**

votes

**1**answer

178 views

### Examples of perfect pseudo algebraically closed fields in positive characteristic

Is there any known example of a perfect pseudo algebraically closed field of positive characteristic containing $\overline{\mathbb{F}_p}$ but is not algebraically closed?

**2**

votes

**2**answers

188 views

### Globally generated, nef and big line bundles which are not ample on a K3 surface

Let $X$ be a K3 surface over $\mathbb{C}$. On a $K3$ surface we know that $Pic(X)\cong Num(X)\cong NS(X)$. A class $L\in Num(X)$ is called movable if $L.C\geq 0$ for every curve $C$ in $X$. It just ...

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**0**answers

94 views

### Stability of automorphism group of complex manifolds

Are there any stability theorems of analytic automorphism groups concerning the deformation of complex manifolds.
For example, in the case of K3 or Calabi-Yau.

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vote

**1**answer

131 views

### Is an Isomorphism from an Abelian variety to a Shimura variety always defined over a solvable extension?

Are there counterexamples to the following:
Given two varieties $A$, $\tilde{A}$, both defined over $\mathbb{Q}$, one of which, say $A$, is a Shimura variety. Then, every isomorphism defined over ...

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votes

**0**answers

237 views

### Vanishing theorems in algebraic geometry and representation theory

Garland proved vanishing theorems for the cohomology of a discrete subgroup with coefficients in a finite dimensional complex representation. As I understand it, Casselman reproved them using the ...

**8**

votes

**1**answer

221 views

### Intuitive reasons for the existence of modular parametrizations

Whenever I encounter anything about modular parametrizations, I have a feeling it is something very unnatural: you have some kind of moduli space and all of a sudden it parametrizes an object ...

**3**

votes

**0**answers

130 views

### Spectral sequences and hypercohomology for projective space

Suppose we are given a complex of sheaves on $\mathbb P^N$ in which every term is direct sum of invertible sheaves:
$$
\mathcal F^\bullet = \dots \to \oplus_{j=1}^{n_{p-1}} \mathcal O (k_j^{p-1}) ...

**19**

votes

**1**answer

631 views

### Lemma 2 from Beilinson's “Coherent Sheaves on $\mathbb{P}^n$ and Problems of Linear Algebra”, intuition?

This is a followup to here.
Consider Lemma 2 from Beilinson's paper "Coherent Sheaves on $\mathbb{P}^n$ and Problems of Linear Algebra", as follows.
Lemma 2. For any pair $i$, $j$ such that $0 ...

**3**

votes

**1**answer

160 views

### In which category does Illusies derived deRham complex live?

In "Complexe cotangent et déformations II", Illusie introduces the derived deRham complex as the pro-completion of the total complex associated to the double complex
$$\Omega_{P_j^A(B)/A}^i$$
where ...

**4**

votes

**1**answer

129 views

### A moduli problem inspired by Stein factorization

Let $f:X \to Y$ be a proper, birational morphism with connected fibers, $X$ is non-singular and $Y$ is normal. Does there exist a moduli space parametrizing all invertible sheaves $\mathcal{L}$ on $X$ ...

**4**

votes

**1**answer

112 views

### Vector bundles with symmetric perfect form

Let $X$ be a smooth projective curve, and $E$ a vector bundle on $X$ such that there exist a bilinear perfect symmetric form $$E\otimes E\rightarrow \mathcal O_X$$
When I see $E$ as a $GL_r$ ...

**6**

votes

**0**answers

92 views

### $K_0$ an $KH_0$ of a normal crossing variety

Let $k$ be a field (say, algebraically closed to fix the ideas) and let $X$ be a strict (aka simple) normal crossing variety over $k$, so that $X$ is union of regular varieties with intersection that ...

**6**

votes

**1**answer

238 views

### Why do we need ampleness in the definition of stability/semistability

This is a general question that I have. Let $X$ be a projective variety over an algebraically closed field $k$. Let $L$ be an ample line bundle over $X$. Let $F$ be a vector bundle on $X$. We say that ...

**4**

votes

**0**answers

141 views

### Vanishing theorems on toric DM stacks

In chapter 9 of the book Toric varieties by Cox-Little-Schenck several cohomology vanishing theorems for toric varieties are proved or mentioned.
In this question I am interested in references for ...

**5**

votes

**0**answers

111 views

### Mirror Symmetry for Homogeneous Spaces other than Flag Manifolds

Mirror symmetry is (reasonably) well understood for the general flag manifolds, due to the work of Kim, Givental, Rietsch, and others. Do there exist other homogeneous spaces for which mirror symmetry ...

**2**

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**0**answers

108 views

### Specialization theorem for cohomology groups

If $f:X\longrightarrow Y$ is a proper lisse morphism and $\mathcal{F}$ is a torsion sheaf on $X$ then one has:
$$
(R^{i}f_{*}\mathcal{F})_{\overline{y}}\cong ...

**1**

vote

**0**answers

99 views

### A naive question on rational equivalence of varieties

Let $X$ be a projective scheme and $\pi:\mathcal{Z} \to \mathbb{P}^1$ a surjective morphism of finite type such that for any pair $t_0, t_1 \in \mathbb{P}^1$, the fibers $\mathcal{Z}_{t_0}$ and ...

**3**

votes

**1**answer

349 views

### Is every complex rational algebraic variety simply connected for the Euclidean topology?

Is it true that every quasi-projective rational irreducible algebraic complex variety is simply connected for the Euclidean topology?
Of course, this is false if we replace "complex" with "real" or ...

**7**

votes

**2**answers

416 views

### Galois cohomologies of an elliptic curve

I asked this question at math stackexchange but did not get any answer and I was suggested to post the question here.
I am studying basic theory of elliptic curves. I encountered Galois cohomology. ...

**14**

votes

**1**answer

648 views

### What's wrong with my understanding of the scheme $\text{Isom}(E_\lambda, E_{\lambda'})$?

Let $\mathcal{M}_{1,1}$ be the moduli stack of elliptic curves (over the complex numbers). Define
$$\begin{eqnarray*} X &:=& \Bbb{A}^1_{\lambda} - \{0,1\}\\
X' &=& \Bbb{A}^1_{\lambda'} ...

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votes

**0**answers

137 views

### Application of Frobenius splitting in characteristic $0$

In general, Frobenius splitting only defines on field of characteristic $p$ (algebraically closed) field.
I am reading Brion and Kumar's book and I can see that there are geometric results can be ...

**2**

votes

**1**answer

111 views

### Pullback of line bundles and divisors from $Kum(C)$ to $Jac(C)$

Let $C$ be a genus 2 curve over $\mathbb{C}$. Let $X=J(C)$. Consider the involution $i$ on $X$, $x\mapsto -x$. Let $Y=\frac{X}{(i)}$. This is a singular surface with 16 points of singularity - these ...

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votes

**1**answer

554 views

### Why are integer points on elliptic curves interesting and useful?

I read some papers which dealed with integer points on elliptic curves. One of these papers are
http://projecteuclid.org/euclid.rmjm/1214947612.
My question is: Why are integer points on elliptic ...

**3**

votes

**1**answer

104 views

### On an example by Romagny about fixed point stack not commuting with coarse moduli space

This is to understand better Example 3.9 on page 221 of Group actions on stacks and applications by M.Romagny.
For an action of an algebraic group (scheme) $G$ on an algebraic stack $\mathcal{M}$, ...

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votes

**1**answer

242 views

### Zero scheme of global sections of vector bundles on affine varieties

I want to understand better the notion of zero scheme of a section of a vector bundle. For simplicity I will consider the case of affine varieties.
Let $\mathbb{K}$ be an algebraically closed field, ...

**3**

votes

**0**answers

111 views

### Direct image of the relative dualizing sheaf

Let $f: X \to Y$ be a flat Cohen-Macaulay (CM) morphism with pure relative dimension $n$ between two projective varieties $X$ and $Y$. Then the relative dualizing sheaf $\omega_{X/Y}$ is $Y$-flat. My ...

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**0**answers

135 views

### Is the Veronese variety “enough” to describe all the $SL(V)$-orbits in $\mathbb{P}(\textrm{Sym}^dV)$?

I apologise in advance if the question will look ridicolous to experienced eyes: in this case a good reference will be enough to clarify my doubts.
Let $V$ be a complex vector space of dimension $n$, ...

**5**

votes

**1**answer

145 views

### curve over higher dimensional basis with 0-dimensional locus of bad reduction

Is there an example of a flat proper relative curve $X/S$ with geometrically connected fibres and with $\mathrm{dim} S > 1$ and $S$ regular and connected with $0$-dimensional locus of bad reduction ...

**2**

votes

**0**answers

93 views

### Resolution of a birational map to get an isomorphism

Suppose that $f:S\rightarrow V$ is a regular map, where S is a compact complex surfaces and $V$ is an algebraic surface such that $f$ induces an isomorphism on the rational function field.
Under ...

**11**

votes

**2**answers

316 views

### Quotient rule, differential operator on a localization is well-defined, underlying geometry?

Using the quotient rule, we obtain that the notion of differential operator on a localization is well-defined:$$\mathcal{D}_A(B_f) \cong \mathcal{D}_A(B)_f.$$Here, $B$ is a commutative $A$-algebra, ...