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

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50 views

### Rationality of a certain real algebraic variety

Let $A_n$ denote the vector space of $n\times n$ antisymmetric matrices over ${\mathbb{Q}}$, where $n$ is even.
Let $A_n^*\subset A_n$ denote the affine ${\mathbb{Q}}$-subvariety of invertible ...

**0**

votes

**0**answers

24 views

### Milnor numbers and mixed multiplicities

section 6 of the link
Teissier showed that Milnor numbers of a hypersurface $(X,0)$ with isolated singulraity at 0 is same as mixed multiplicities of the Hilbert polynomial of the filtration ...

**0**

votes

**0**answers

41 views

### $z^n-t^m=x^3+y^3$ and Vojta's more general abc conjecture

In A more general abc conjecture, p. 7 Paul Vojta conjectures:
If $x_0,\ldots x_{n-1}$ are nonzero coprime integers satisfing $x_0 + \cdots x_{n-1}=0$
$$ \max\{|x_0|,\ldots |x_{n-1}|\} \le C ...

**1**

vote

**0**answers

104 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 ...

**0**

votes

**1**answer

165 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 ...

**4**

votes

**1**answer

150 views

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

I am having 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 ...

**7**

votes

**0**answers

115 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 ...

**4**

votes

**1**answer

139 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, ...

**1**

vote

**0**answers

86 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 ...

**4**

votes

**2**answers

235 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 ...

**0**

votes

**0**answers

37 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 ...

**0**

votes

**0**answers

86 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 ${\mathbb CP}^1$ ...

**2**

votes

**1**answer

151 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 ...

**1**

vote

**1**answer

160 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 ...

**6**

votes

**1**answer

371 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 ...

**5**

votes

**0**answers

188 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 ...

**3**

votes

**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 ...

**7**

votes

**0**answers

191 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 ...

**1**

vote

**0**answers

111 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 ...

**1**

vote

**1**answer

249 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 ...

**1**

vote

**0**answers

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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votes

**0**answers

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 ...

**-2**

votes

**0**answers

133 views

### Reference for algebraic manipulation of sheaves [closed]

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 ...

**2**

votes

**1**answer

83 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 ...

**3**

votes

**1**answer

228 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 ...

**1**

vote

**0**answers

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 ...

**1**

vote

**1**answer

152 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 ...

**6**

votes

**1**answer

854 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? ...

**4**

votes

**0**answers

171 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
...

**0**

votes

**0**answers

188 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 ...

**1**

vote

**0**answers

75 views

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

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 ...

**3**

votes

**0**answers

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 ...

**5**

votes

**0**answers

125 views

### Symplectic invariance of Hodge numbers?

Let $(X,\omega)$ be a compact symplectic manifold. If $J$ is a $\omega$-compatible complex structure on $X$ then $(X,\omega,J)$ is a compact Kähler manifold and so has Hodge numbers $h^{p,q}$.
My ...

**2**

votes

**2**answers

258 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, ...

**4**

votes

**1**answer

287 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 ...

**2**

votes

**1**answer

182 views

### Number of rational points in a non-smooth variety

Let $X$ be an algebraic variety over $\mathbb{F}_q$ with dimensional $n$. We know that if $X$ is smooth than $X$ has about $q^{nk}$ rational points over $\mathbb{F}_{q^k}$ (Weil hypothesis). Is there ...

**2**

votes

**1**answer

218 views

### Using Lefschetz duality in algebraic geometry

I am reading the paper of Fulton and Lazarsfeld on the connectivity of degeneracy loci of morphisms of vector bundles, but there is a comment in the article that I don't quite understand.
Let $G$ be ...

**0**

votes

**0**answers

155 views

### Exact sequence of vector bundles

Consider the short exact sequences below;
\begin{equation}
0\longrightarrow H^0(\mathbb{P}^4,\mathcal{O}_{\mathbb{P}^4}(d-1)^{\oplus 4})\longrightarrow ...

**4**

votes

**1**answer

154 views

### Smooth real points on the intersection of a quadric and a cubic

Let $C$, $Q \in \mathbb{R}[x_0,\dots,x_n]$ be homogeneous of degrees $3$ and $2$ respectively. Consider the scheme $V$ in $\mathbb{P}^n$ defined by
$$ V \; : \; C=Q=0$$.
Suppose
$V$ is integral ...

**1**

vote

**1**answer

173 views

### Irreducible variety

I asked a similar question at MSE, as the question seemed quite basic to me, but did not get any hint in 24 hours, except for one upvote for the question itself. I still think I am stuck with some ...

**5**

votes

**1**answer

183 views

### Cotangent complex of certain dg-scheme

This is a somewhat embarrassing question, but still I will ask it. Let $V$ be a vector space over $\mathbb C$ of dimension $d$.
Let $X$ be the dg-preimage of $0$ under the natural map $V\to Sym^2(V)$ ...

**1**

vote

**0**answers

102 views

### Calabi-Yau with nodes

Suppose $X$ is a singular projective irreducible complex variety of dimension 3, and its singular loci are finite number of nodes, and its smooth locus $X_1$ is a Calabi-Yau quasi-projective variety, ...

**4**

votes

**0**answers

150 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 ...

**5**

votes

**2**answers

558 views

### A transcendence question involving the exponential function

Let $(z_n)$ be a sequence of complex numbers satisfying $|z_n|\to +\infty$ and such that $\{e^{z_n}\mid n \in \mathbb{N}\}$ is infinite.
Is it always true that $\{(z_n,e^{z_n})\mid n \in\mathbb{N}\}$ ...

**6**

votes

**1**answer

154 views

### If a faithfully flat extension of dg/A_$\infty$-algebra is formal, is the original algebra formal (over positive characteristic)?

Proposition 6.2 of Formality of DG algebras (after Kaledin) by Lunts reads (with a few additions to clarify notation):
Let $k$ be a field of characteristic 0. Let $A$ be an $A_\infty$ algebra ...

**6**

votes

**1**answer

124 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) ...

**2**

votes

**0**answers

94 views

### Quantizable vs. integral Kahler form

Let $(M,\omega)$ be a (not necessarily compact) Kahler manifold. Then the form $\omega$ is integral if and only if $\omega \in c_1 (L) $ for some holomorphic line bundle $L$.
A Hermitian holomorphic ...

**3**

votes

**0**answers

343 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 ...

**2**

votes

**0**answers

98 views

### local description of $\mathbb{P}^2$-fibrations over $\mathbb{P}^1$

Let $X$ be a rational threefold (over the field of complex numbers) with terminal singularities. It is well-known that $X$ has only finitely many singular points $x_1,x_2, \ldots,x_n$.
To be more ...

**0**

votes

**0**answers

67 views

### Do closed points have “locally maximal” codimensions?

Let $S$ be a Noetherian excellent irreducible scheme of finite Krull dimension; since the question is local it may also be assumed to be affine. Let $s$ be a closed point of $S$. My question is: does ...