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

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127 views

### Field extension and nilpotent element

Let $k$ be an algebraically closed field of characteristic $p>0$, $A$ a regular local noetherian $k$-algebra, $B$ another local noetherian $k$-algebra, $f:A \to B$ an injective ring homomorphism of ...

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50 views

### Base change and geometrically generic reduced fiber

Let $k$ be an algebraically closed field of characteristic $p>0$ and $f:X \to Y$ be a quasi-projective morphism between noetherian $k$-schemes. Assume that $Y$ is regular and the geometric generic ...

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**1**answer

149 views

### Monodromy theorem of degeneration of smooth projective varieties to non-reduced central fiber

Given a family $\pi: \mathcal{X}\rightarrow\Delta$ smooth away from $0\in\Delta$, Where $\mathcal{X}$ is a smooth complex manifold, $\Delta$ is a small disk, the general fiber of $\pi$ is smooth ...

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86 views

### A criterion for orbits of complex reductive group to be closed

I have some trouble understanding Nakajima's proof of the Kempf-Ness theorem in [1]. At the end (proof of Proposition 3.9(6)), his argument is basically the following:
Let $G=K_{\Bbb C}$ be a ...

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**1**answer

130 views

### Schwartz-Zippel lemma for an algebraic variety

Let $X $ be a smooth affine subvariety of $(\overline{\mathbb{F}_q})^n$ defined by a prime ideal $I$. Let $f$ $\in \mathbb{F}_q[x_1,\ldots,x_n]$ be a polynomial such that $f \notin I$.
Let $r_1, ...

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98 views

### Are there genera for algebraic cobordism?

For real and complex manifolds, we can form the (oriented) cobordism ring $\Omega$, and a genus is defined to be a ring homomorphism
$$\varphi:\Omega\otimes\mathbb{Q}\to R$$
where $R$ is any ...

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**1**answer

880 views

### Euler Characteristic of a Variety

Let $Y$ be a "nice" scheme. I am thinking projective varieties over an algebraically closed field, for now, but I am open to more general results.
In terms of singular homology ...

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218 views

### Families of abelian varieties on the line (or more generally simply connected varieties)

I'm curious whether the following is true:
Question 1: Let $V/\mathbb{C}$ be a smooth connected variety such that $V^\text{an}$ is simply connected. Then, is every abelian scheme $f:\mathscr{A}\to ...

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76 views

### Smoothness of a (given) global section of a vector bundle over G(2,6)

Let $G=Gr(2,6)$ the Grassmannian of two planes in $V=\mathbb C^6$, and let $\mathcal Q(1)$ the rank four quotient bundle on it twisted with $\mathcal O_G(1) \cong $ det$(S^*)$, $S$ being the ...

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**1**answer

122 views

### Conditions for a smooth scheme of finite type with trivial class group to be quasi-affine

Let $X$ be a smooth scheme of finite type over an algebraically closed field of characteristic zero and with a trivial class group $Cl(X)=0$. Let $Y$ be a dense open subscheme of $X$ such that:
1) ...

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**1**answer

364 views

### what is the universal cover of GL(2,R)?

In the theory of Bridgeland stability conditions one has an action of the universal cover $G'$ of $G = GL^+(2,\mathbb R)$.
What is G'?
I know there is concrete description in terms of pairs ...

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**1**answer

141 views

### Which varieties are flat degenerations of projective space?

Let $V$ be a vector space over a field with discrete valuation and let $R$ be its ring of integers.
Which varieties can be reached as the special fiber of a flat degeneration over $R$, when the ...

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**1**answer

75 views

### Sections of a linear system splitting as a product of degree one polynomials

Let $X\subset\mathbb{P}^n$ be a hypersurface of degree $d$ and with multiplicities $m_1,...,m_k$ at $p_1,...,p_k\in\mathbb{P}^n$ general points.
Let $S\subseteq |\mathcal{O}_{\mathbb{P}^n}(d)|$ be ...

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187 views

### Morphisms for good reduction are maps respecting filtration

Please see edit below.
So, let $A,A'/K$ be abelian varieties where $K$ is a $p$-adic local field with residue field $k$. Suppose further that they have good reduction with models ...

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35 views

### Finding functions on curves with given multiplicity at a point

just migrated these questions from SE, I hope is ok.
Let $C/k$ be a smooth curve over a field (not necessarily perfect) , $P\in C(k)$ and $n\in \mathbb{Z}^{+}$ such that $n>1$
I want to find a ...

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81 views

### Does there exist an Affinization or Projectivization process for Varieties?

Let us consider the classical isomorphism of real manifolds between $S^2$ and ${\mathbb CP}^1$. First strange thing we have here is that both are varieties, but $S^2$ is an affine and ${\mathbb CP}^1$ ...

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**1**answer

372 views

### Equidimensionality of stalks of $\operatorname{Proj} S$ when $S$ is equidimensional.

I would like to know a reference of the following statement (or counter example).
Let $S$ be a (commutative) Noetherian standard graded ring over a local ring, i.e., $S = S_0[S_1]$, where $S_0$ is ...

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**1**answer

849 views

### How much of modern algebraic geometry is there in modern complex(algebraic, analytic, differential) geometry?

Good day to you, people of mathoverflow. I'll get to the point. I wonder how much of modern(abstract) algebraic geometry is there in modern complex geometry?
What do I mean by complex geometry? ...

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181 views

### Deligne's theorem on the characterisation of Tannakian categories

I am trying to understand the proof of the Theorem 7.1 from Catégories Tannakiennes by Pierre Deligne https://publications.ias.edu/sites/default/files/60_categoriestanna.pdf.
Essentially, it is ...

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148 views

### Some examples where the plurigenera are nonconstant, when the fibres have worse singularities than canonical

Let start with a definition
Invariance of plurigenera: Choose $m$ large enough so that $mK_F$ has a non-zero global section for some fibre $F$. For any fibre $F$, we have $K_F = K_{X/D}~_{|F}$. So ...

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374 views

### Are ranks of Jacobians over number fields unbounded?

Fix a number field $K$.
Is the rank of $J(K)$ unbounded, where $J$ ranges over the Jacobians of all smooth, projective, geometrically connected curves over $K$?
Does there exist an integer $g$ such ...

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202 views

### endomorphisms of the Jacobian of a curve

Let $C$ be a smooth, projective curve of genus >1 over the complex numbers and let $J(C)$ be its Jacobian. The Torelli theorem relates the automorphisms of $C$ to the automorphisms of $J(C)$. ...

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151 views

### General Reference for surface singularities

Is there any "standard" reference for (rational) singularities on algebraic surfaces? I'm aware of Artin's papers and the one of Brieskorn (Rationale Singularitäten komplexer Flächen), but they seem ...

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132 views

### Reference for algebraic manipulation of sheaves [on hold]

I am currently playing with sheaves over families of algebraic varieties ($O_x$-modules) their torsion sub-sheaf, higher direct images and tensor products, I am looking for a good reference to learn ...

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**1**answer

171 views

### Maximal length of filter regular sequence

Let $A$ be a Noetherian ring and $R$ is a standard graded ring over $A.$ Let $M$ be a finitely generated graded $R$-module and $I$ be a graded ideal of $R.$ Then $x_1,\ldots,x_r\in I$ is called ...

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**1**answer

284 views

### an algebraic variety for a boolean circuit

There is a polynomial reduction from a $3-CNF$ $SAT$ problem to some system of polynomial equations over $\mathbb{F}_2$.
I mean there is polynomial reduction $F$ such that for every boolean ...

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**3**answers

918 views

### moduli spaces are kahler?

I often heard from experts that "moduli spaces are Kahler". This sounds as a meta-theorem asserting that every time one defines reasonable moduli spaces, then there is a standard strategy to see ...

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341 views

### Finding the number of rational points effectively

Consider $\# P$ and $\oplus P$.
There is a $\# P$-hard problem: to find number of rational solutions of a system of polynomial equations over $\mathbb{F}_2$. The corresponding $\oplus P$-hard ...

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110 views

### Isomorphism vs. projective equivalence: the $10$-dimensional spinor variety

Let $S$ be the $10$-dimensional Spinor variety parametrizing one of the two families of $4$-dimensional linear subspaces of the non-singular quadric in $\mathbb{P}^{9}$. I have read that there exist ...

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68 views

### Connectedness of fibers of a map related to the secant variety

Let $X\subsetneq\mathbb{P}^{N}$ be a smooth projective variety, and let
$$
S_{X}=\overline{\{(x,y,z)\in X\times X\times \mathbb{P}^{N}:x\neq y, z\in\langle x,y\rangle\}}.
$$
The secant variety of $X$ ...

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248 views

### Automorphisms of rings fixing all prime ideals

Let $f,g:A \to B$ be two ring homomorphisms of noetherian rings satisfying that for any prime ideal $\mathfrak{q} \subset B$, $f^{-1}(\mathfrak{q})=g^{-1}(\mathfrak{q})=:\mathfrak{p}$ and the induced ...

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275 views

### Which bundles does the character variety parameterize?

For any Riemann surface with punctures $C$, and Lie group $G$, the character variety is the space of maps $\mathrm{Hom}(\pi_1(C), G)$.
I know that if $G= S_n$ (not a lie group), then ...

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58 views

### Need a reference/proof for computing the regularity of ideal of points in $\mathbb P^d$?

In a lecture notes on 'Cohomology modules' i read the following remark:
Given a set $X$ of points in $\mathbb P^d$,using the Local Cohomology modules one can easily compute the reg$(S_X)$ where ...

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68 views

### Question regarding Geometric meaning of Noether normalization theorem for projective varieties [migrated]

In Ernst Kunz's ''commutative algebra and algebraic geometry'' book, ch.2, proposition 4.5 the author states:
The Noether Normalization Theorem admits the following application in projective ...

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**1**answer

303 views

### Grothendieck class and Normalization

Does the Grothendieck class behave well under normalization? Here by Grothendieck class, I mean the class of a variety in $K_0(\mathcal{var/\mathbb{C}})$

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74 views

### Rank diagrams of permutations $w \in S_{m}$ in the study of complete flag varieties [on hold]

I'm looking for some good references that may either prove or help to prove the following statement: Show that a matrix $r=(r_{pq})_{1 \leq p,q \leq m}$ defines a rank diagram for some pair of ...

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**1**answer

413 views

### Character varieties of finitely generated groups

Consider the following situation: $\Gamma_0\leq\Gamma$ are both finitely generated groups and $\Gamma_0$ has finite index in $\Gamma$. The restriction gives a well defined map between the character ...

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**1**answer

81 views

### On some curves of real values of a rational function

For given parameters $a_{1},\dots,a_{k}\in\mathbb{R}$, define the rational function $\phi:\mathbb{C}\to\mathbb{C}$ as
$$\phi(z)=\frac{1}{z}-a_{1}z-a_{2}z^{2}-\dots-a_{k}z^{k}.$$
The domain of its real ...

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**1**answer

226 views

### Euler characteristic - reference question

Let $X$ be an algebraic variety over $\mathbb C$ and let $\mathcal F$ be a constructible sheaf on $X$. It is well-known that the Euler characteristic of the cohomology of $\mathcal F$ is equal to the ...

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**1**answer

353 views

### Can monodromy be described by the same matrix for chosen generators in case of the same singularity type?

Let $X$ be a surface in $\mathbb{P}^3$. We have a fibration $f: X \longrightarrow \mathbb{P}^1$, and $f^{-1}(s_1)$ and $f^{-1}(s_2)$ have the same singularity type. Let $\gamma_1$ and $\gamma_2$ be ...

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441 views

### is the modular curve X(N) defined over Q?

In most sources, the field of definition of the modular curve $X(n)_\mathbb{C}$ (quotient of the upper half plane by the subgroup $\Gamma(n)$ of $SL_2(\mathbb{Z})$ congruent to $I\mod n$) is ...

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187 views

### On the coherence of formal power series ring

Let $A = {\Bbb F}_p[[X_1,X_2,...]]$
be the ring of formal power series with infinitely many variables over the finite field ${\Bbb F}_p.$
$A$ consists of such formal sum elements as $\sum ...

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**1**answer

523 views

### What sort of ind-scheme is this?

It apparently follows from work of Velu (MathSciNet) that every isogeny between elliptic curves in (long) Weierstrass form over $k$ can be written in the form
$$
\left(\frac{u(x)}{v(x)}, ...

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129 views

### Rationally connected spaces over non-algebraically-closed fields

The definition I most often see for what it means for a projective variety $X$ over a field $k$ to be rationally connected is that there exists a variety $M$ and a dominant morphism ...

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326 views

### Generalization of the rigidity lemma in birational geometry

Let $X,Y,Z$ be projective varieties, and let $f:X\rightarrow Y$, $g:X\rightarrow Z$ be dominant morphisms. Assume that all the fibers of $g$ have the same dimension and are connected.
If there exists ...

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170 views

### A suspected typo, and Deligne's image of the general fiber swallowing the special

In SGA 4.5 (Arcata V.1) Deligne writes:
Let $X$ be a complex analytic variety and $f:X\rightarrow D$ map $X$ into the
disk. Write $[0,t]$ for closed line segment with extremities 0 and $t$ in
...

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188 views

### Matsushita theorem on framed variety (X,D)

I have a question about fibrations on Irreducible log holomorphic symplectic manifolds. Lets give some introduction
Motivation; A holomorphic symplectic manifold (HSM) is a $2n$-dimensional compact ...

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48 views

### Efficient CW structures on squarefree semi-algebraic set

General Setup
Given a collection of $k$ polynomials (with real coefficients) in $n$ real variables, say $f_i(x_1,\ldots,x_n)$, let $V \subset \mathbb{R}^n$ correspond to those $x$-values for which ...

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257 views

### Irreducible algebraic sets via irreducible polynomials

There are many results about irreducible polynomials over finite fields:
we know a cardinality of all irreducible polynomials with given degree, we know explicit examples of irreducible polynomials, ...

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202 views

### How to check that an ideal of $\mathbb{C}[GL_n]$ is a coideal or not?

Let $I$ be an ideal of $\mathbb{C}[GL_n]$. Are there effective methods or software to check whether $I$ is a coideal or not? Thank you very much.
For example, let I be the ideal of $\mathbb{C}[GL_3]$ ...