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

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**1**answer

76 views

### pull-back of canonical divisor under blow-up of a singular point

I was checking an example of canonical singularities from surface.
We consider the surface $X:(xz=y^2)\subset \mathbb A^3$. The only singular point is the origin. We write down one affine piece of ...

**2**

votes

**1**answer

160 views

### What is the cohomology of the tangent bundle of a flag variety?

Let $G$ be the general linear group $\operatorname{GL}(n,\mathbb{C})$ and $P$ a parabolic subgroup with Lie algebra $\mathfrak{p}$. Consider the vector bundles
$$
\mathcal{P} = G\times_P ...

**2**

votes

**4**answers

354 views

### applications of Berkovich spaces

What are applications of the theory of Berkovich analytic spaces? The analytification $X \mapsto X^{\mathrm{an}}$

**4**

votes

**1**answer

260 views

### Covering of schemes and flatness

Let $f:X \to Y$ be a finite surjective morphism of quasi-projective schemes over $\mathbb{C}$, $X$ is reduced and $Y$ is integral. Suppose that there exists an integer $n$ such that for every closed ...

**0**

votes

**0**answers

132 views

### For which pairs of distinct positive primes $p$ and $q$, the integral closure of $\mathbb{Z}$ in $\mathbb{Q}[\sqrt{pq}]$ is a UFD?

For which pairs of distinct positive primes $p$ and $q$, the integral closure of $\mathbb{Z}$ in $\mathbb{Q}[\sqrt{pq}]$ is a UFD? I've proved that neither $p$ nor $q$ can be congruent to $1$ modulo ...

**0**

votes

**1**answer

178 views

### Reference Request: Fundamental Group Scheme

I want to learn about the Fundamental Group Scheme(First introduced by Madhav Nori). I am familiar with Basic Algebraic Geometry at the level of Eisenbud & Harris'"Geometry of Schemes" & to a ...

**1**

vote

**0**answers

103 views

### Intersection Multiplicity

Let $X$ be an hyper-surface in an affine space defined by an equation $F$. We can assume that the ground field is $\mathbb{C}$ and $X$ is normal. Take functions $f_1,\dots, f_n$ on $X$ and let $B$ ...

**4**

votes

**1**answer

72 views

### Estimate self crossings of a curve parameterized by a trigonometric polynomial

Given z on the unit circle, let $P(z)= \sum\limits_{k=-D}^D p_k z^k $.
Can one estimate the number of self crossings of the following curve with an analytic expression in terms of the coefficients ...

**5**

votes

**0**answers

145 views

### Does the fundamental group identify group structure on subvarieties of products of curves?

Let $C_1,\dots, C_n$ be smooth curves over $\overline{\mathbb F}_p$, not necessarily proper. Let $X$ be a subvariety of $C_1 \times \dots \times C_n$. I'm interested in the natural map:
$$ ...

**3**

votes

**1**answer

170 views

### Analytically but not algebraically smoothable singularity

Are there examples of algebraic singularities which may be smoothed analytically but not algebraically? It certainly seems possible, but if not, why? Are there conditions under which this becomes ...

**1**

vote

**0**answers

68 views

### miniversality vs versality

Consider a moduli problem $\mathcal{M}$. Assume, at each point $x$, the associated deformation problem $\mathcal{M}_x$ has a tangent-obstruction theory. It follows that $\mathcal{M}_x$ has a hull ...

**2**

votes

**1**answer

130 views

### A question on polynomial heights

For a given polynomial $f(x_1, \cdots, x_n) \in \mathbb{Z}[x_1, \cdots, x_n]$, define the height of $f$ as $H(f)$ as the maximum absolute value among its coefficients. We can also define the log ...

**1**

vote

**1**answer

103 views

### Degree of irreducible locally free sheaves and global sections on curves

Let $X$ be a smooth projective curve and $\mathcal{F}$ a locally free sheaf on $X$ of rank $2$ and negative degree. Assume further that $\mathcal{F}$ is irreducible in the sense, $\mathcal{F}$ cannot ...

**0**

votes

**1**answer

194 views

### Find all possible rational values of the parameter of a parametric cubic such that it is reducible

Description: Given the following parametric cubic polynomials ${E}^{3}
- 15\, {\beta}_{\pm}\, {E}^{2} - 3 \left({71\, {\beta}_{\pm}^{2} + 352\, {\beta}_{\mp}}\right) E
+ 135\, {\beta}_{\pm} ...

**0**

votes

**2**answers

126 views

### divisor is big iff its birational pullback is big

In Kollár and Mori's Birational Geometry of Algebraic Varieties, the authors say a Cartier divisor is big iff its birational pullback is big (Definition 2.59 below). But I can't understand. Maybe the ...

**1**

vote

**1**answer

138 views

### Special linear sections of a hypersurface

We discuss on the field of complex numbers. Let $X$ be a smooth projective hypersurface of dimension $n \geq 4$ in $\mathbb{P}^{n+1}(\mathbb{C})$. Assume that for a general point $x \in X$, there ...

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votes

**0**answers

54 views

### About bounding values of quadratic forms

It would be helpful if someone can share (either as references) examples of calculations/analysis which achieves bounding of values of quadratic forms in say either of the following situations,
...

**4**

votes

**2**answers

193 views

### Fixed space of the square of a symmetric matrix over $\mathbb{F}_2$

Let $M$ be an invertible symmetric $2n \times 2n$ matrix with entries in the finite field $\mathbb{F}_2$. Is $\mathrm{Ker}\ (M^2 - I_{2n})$ necessarily even dimensional?

**1**

vote

**1**answer

188 views

### Pushforward of pullback

Let $f \colon X \to Y$ be a (proper if you want) morphism between two noetherian schemes. Let $X_0$ and $Y_0$ be two closed subschemes of $X$ and $Y$, with closed immersion $i \colon X_0 \to X$ and $j ...

**4**

votes

**0**answers

176 views

### Fundamental group of the complement homogeneous variety in $\mathbb{C}P^{n-1}$

Let $f,g:\mathbb{C}^n\to \mathbb{C}$ are two irreducible homogeneous polynomials. If there is a homeomorphism $h:\mathbb{C}^n\to \mathbb{C}^n$ such that $h(X)=Y$ and $h(0)=0$, where $X=f^{-1}(0)$ and ...

**2**

votes

**1**answer

133 views

### integral basis for the Lie algebra of the Neron model of an abelian variety

Let $A$ be an abelian variety over a number field $K$. Let $\mathcal{A}$ be the Neron model of $A$ over $O_K$. Let $\Omega_{\mathcal{A}/O_K}$ be the sheaf of invariant differential forms on ...

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votes

**0**answers

171 views

### What is the meaning of “field of constants”?

Let $X$ be a smooth proper variety over a field $k$, and let $D \subset X$ be a smooth irreducible divisor.
What is the meaning of "the field of constants of $D$"?
Here is my guess: $D$ is proper, ...

**1**

vote

**0**answers

141 views

### How to prove that any perfect complex on an affine scheme is strictly perfect?

Let $(X,\mathcal{A})$ be a ringed space. A complex $\mathcal{S}^{\bullet}$ of $\mathcal{A}$-modules is $\textit{perfect}$ if for any point $x\in X$, there exists an open neighborhood $U$ of $x$ and a ...

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votes

**0**answers

117 views

### Rational group scheme

Suppose $G$ is a group scheme over a field $k$, i.e., $G$ is a functor from the category $\text{Alg}_k$ of unital commutative, associative $k$-algebras to the category of $\text{Groups}$. Suppose that ...

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votes

**0**answers

89 views

### Local cohomology, vanishing of cohomology for sheaves that are not $\mathcal{O}_X$-modules

Let $X$ be a scheme over a field $k$ and x$\in X$ a closed point. Then one can calculate $H^1_x(X,\mathcal{O}_X)$ to be isomorphic to $\mathcal{O}_{X,x}[1/f]/\mathcal{O}_{X,x}$ using the exact ...

**5**

votes

**0**answers

116 views

### Gersten Conjecture for Milnor K-theory

The Gersten conjecture for Milnor K-theory, saying that the Gersten complex $$0\rightarrow \mathcal{K}^M_X\rightarrow \oplus_{x\in X^0}i_{x*}(K^M_n(x))\rightarrow \oplus_{x\in ...

**1**

vote

**1**answer

175 views

### If X is a quasiprojective variety with condition S_2, and if its normalization Y is Cohen-Macaulay, is X necessarily Cohen-Macaulay?

Suppose that $X$ is a quasiprojective variety satisfying $S_2$, and that the normalization ${X}^{\mathrm{N}}$ of $X$ is Cohen-Macaulay.
Question: Is $X$ Cohen-Macaulay?
This already fails for ...

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votes

**1**answer

235 views

### Is every surjective, birational transformation of projective varieties automatically proper?

Let $X$ and $Y$ be two complex, irreducible, normal, projective varieties (read: integral, projective, normal $\mathbb C$-schemes of finite type), projective in the sense of Hartshorne.
Let ...

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votes

**0**answers

88 views

### Relation between dualization complex, cotangent complex and Deligne-Du Bois complex?

Given a smooth variety $X$, one can define the cotangent sheaf $\Omega_X$, the canonical sheaf $\omega_X$ and the deRham complex $\Omega_X^\bullet$. These three object has obvious relations.
For ...

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votes

**0**answers

64 views

### Terminology regarding divisor on a curve

Suppose that $D = \sum n_i P_i$ is a divisor on a curve $C$, say, over a field. Is there a standard algebraic geometry terminology referring to the set $\{ P_i : n_i \neq 0 \} \subset |C|$? Support of ...

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votes

**3**answers

583 views

### Blow-ups and blow-downs

Can $\mathbb P \times \mathbb P \times \mathbb P$ be obtained from $\mathbb P^3$ by a finite succession of blow-ups and blow-downs along non-singular centers?

**-3**

votes

**0**answers

70 views

### Blow-ups and blow-downs of $\mathbb P^3$ [duplicate]

If one blows up $\mathbb P^2$ twice and blows down once, then he may get $\mathbb P \times \mathbb P$.
I am wondering if the same thing can be done for $\mathbb P^3$ and $\mathbb P \times \mathbb P ...

**5**

votes

**1**answer

161 views

### Constructing Ramanujan graphs from elliptic curves

Is there an exposition which explains how to do this step-by-step? (I see stray references and allusions to such a thing being possible but can't locate anything concretely)
Something to do with ...

**0**

votes

**2**answers

109 views

### Hochster-Roberts Theorem reciprocal

Given a Cohen-Macaulay ring $R$ over a field of characteristic zero and $G$
a reductive algebraic group acting on $R$, then the ring of ivanriants $R^G$
is also Cohen-Macaulay. This is known as ...

**1**

vote

**0**answers

135 views

### Classifying Spaces and Eilenberg-Maclane objects in the category of simplicial rings

[Skip down to the bottom for a correction] Let's work over a field k, assume it is as nice as you need it to be.. Suppose I have an ordinary (edit: commutative) affine group scheme G = Spec(A) over k, ...

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vote

**2**answers

333 views

### Semistability in GIT

If I understand correctly, in geometric invariant theory, polystable points can be defined as those which have a closed orbit. Is it true that semistable points can be characterized as those whose ...

**4**

votes

**0**answers

188 views

### Brauer-Manin obstruction to surfaces of Kodaira dimension 1

Roughly speaking, the Kodaira dimension is an invariant of a variety that corresponds to curvature. One can show that curves of genus $\geq 2$ have Kodaira dimension 1 using Riemann-Roch. In Corollary ...

**3**

votes

**1**answer

146 views

### Periodic points in C^2

I came up with a problem which is similar to the following quesitons:
Consider a map: $f(x,y)=(y^2-2,xy-2)$. It is seems that the number of periodic points of given period is bounded.
I want to ...

**23**

votes

**5**answers

2k views

### Maryam Mirzakhani's works

Maryam Mirzakhani has made several contributions to the theory of moduli spaces of Riemann surfaces.
Mirzakhani was awarded the Fields Medal in 2014 for "her outstanding contributions to the dynamics ...

**16**

votes

**2**answers

403 views

### Variety acquiring rational point over any quadratic extension

Does there exist a variety $X$ over $\mathbb{Q}$ (or a number field) such that it has no rational points over $\mathbb{Q}$ but acquires points over any quadratic extension $\mathbb{Q}(\sqrt{d})$?
If ...

**3**

votes

**1**answer

157 views

### Converse to Weil Restriction of Scalars

Let $k$ be a field of characteristic zero (I'm only interested in number fields), and let $\mathbb{G}_{/k}$ be a linear algebraic group defined over $k$ which is almost $k$-simple (all normal ...

**5**

votes

**0**answers

130 views

### Is the moduli space of curves arising from wild ramification smooth?

Fix a natural number $g$, a prime $p$, and a $p$-group $P$.
Let $C$ be a smooth projective curve of genus $g$ with a faithful action of $P$ and an isomorphism $C / P \cong \mathbb P^1$ such that $P$ ...

**3**

votes

**1**answer

200 views

### Motivic integration in positive characteristic: how much is known?

It seems that in papers on motivic integration people usually assume the base field to have characteristic $0$ (and algebraically closed?). My question is: how much can one prove over a positive ...

**2**

votes

**0**answers

243 views

### When an elliptic curve is a quotient of $\mathbb{G}_a$?

I want to know when an elliptic curve $E \rightarrow S$ is a quotient of $\mathbb{G}_a$.
When $S$ is an analytic space, there is an exact sequence $$0 \rightarrow R^1 \mathbb{Z}^{\vee} \rightarrow ...

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votes

**0**answers

99 views

### Fermat's Theorem on p = a^2 + b^2 [migrated]

I have read that Fermat predicted that for an odd prime $p$, $p = a^2 + b^2$ iff $p = 1$ mod 4.
I heard that such a criterion could be possible for a given integer $n$ like
$p = a^2 + n b^2$
...

**8**

votes

**1**answer

300 views

### A reference for $\mathbb{A}^1_R$ being a coarse moduli space of the stack of elliptic curves

Let $R$ be a ring and let $\mathcal{M}_{1, R}$ be the algebraic $R$-stack of elliptic curves (over $R$-schemes as bases). One knows that the coarse moduli space of $\mathcal{M}_{1, R}$ is supposed to ...

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votes

**1**answer

98 views

### Relation between intersection multiplicities

Consider $(f_1,\dots,f_n), (g_1,\dots,g_n)\in \mathbb{C}[z_1,\dots,z_n]\ $ such that:
i) $\{f_1=\dots=f_n=0\}= \{g_1=\dots=g_n=0\}=\{0\}\in \mathbb{C}^n\ $ and
ii) $f_1g_1+\dots+f_ng_n\equiv0$.
...

**0**

votes

**2**answers

226 views

### Serre's Theorem for Coherent Sheaves

I recently heard a discussion about a certain of Serre which reconstructs the category of coherent sheaves of a variety $V$ as the category of modules over the homogeneous space of $V$ modulo modules ...

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vote

**4**answers

252 views

### Global section of very ample line bundles and its value on stalks

Let $X$ be a projective scheme and $\mathcal{L}$ be a very ample line bundle on $X$ with respect to some projective embedding $X \hookrightarrow \mathbb{P}^n_{\mathbb{C}}$ (for some $n$).
Given any ...

**3**

votes

**1**answer

148 views

### Does the ring of invariants inherit normality?

Let $A$ be a normal ring (in the sense that its localizations at prime ideals are normal domains), and suppose that a finite group $G$ acts on $A$ by ring automorphisms. Form the subring $A^G \subset ...