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

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**1**answer

190 views

### Degree of the negative part of a divisor

Let $K$ be an algebraically closed field (or $\overline{\mathbb{C}(z)}$ for a more precise condition). And let $P \in K[x,y]$ be an irreducible polynomial of degree $m$ with respect to $x$ and degree ...

**5**

votes

**2**answers

518 views

### pullback of theta divisor

Let $C$ be a smooth curve and $J$ its Jacobian. Let $p$ be a point on $C$ and $j: C \to J$ be the map $x \mapsto x-p$. Let $\theta$ be the Theta divisor on $J$, i.e. the locus $\{ x_1 + \cdots + ...

**5**

votes

**0**answers

209 views

### Map of the Klein quartic from $CP^2$ to $R^3$

The Klein quartic $\mathcal{Q}$ is cut out of $\mathbb{CP}^2$ by the homogeneous equation $$x^3 y + y^3 z + z^3 x = 0.$$ It has 168 orientation preserving automorphisms and includes several copies of ...

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votes

**1**answer

119 views

### Vanishing natural transformation and strong generator

Let $X$ be a smooth projective variety (over the field of complex numbers). Let $T$ be strong generator of $D^b(X)$ : this means that every object in $D^b(X)$ can be obtained in a given finite number ...

**5**

votes

**1**answer

756 views

### Recent ideas in Macaulayfication?

Kawasaki has shown that a quasi-projective variety over a field has a Macaulayfication. His construction does not preserve the Cohen-Macaulay locus of the original variety, only a finite set of ...

**5**

votes

**0**answers

120 views

### definition of “immersion” of schemes (without open or closed)

On Prop. 1.7 (a) on page 5 of Milne's Etale Cohomology book states:
Any immersion is quasi-finite.
A google search turned up definitions for "open immersion" and "closed immersion", never just ...

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vote

**0**answers

59 views

### Calculations about the normal bundle of embedding of symmetric products

Suppose $C^{(d)}$ is the $d$-th symmetric product of a curve, embed it in $C^{(d+s)}$ by $E\to E+sP_0$ where $P_0$ is a fixed point. The normal bundle of this embedding is denoted by $N$.
Suppose ...

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**0**answers

99 views

### Commuting diagram, algebraic cycles and K-theory

What is the easiest way to see the veracity of the following commutative diagram?$$\require{AMScd}
\begin{CD}
K(X) \otimes K(Y) @>\text{ch}(-) \otimes \text{ch}(-)>> A(X, \mathbb{Q}) \otimes ...

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vote

**0**answers

71 views

### Rank of the Jacobian of a family of hyperelliptic curves of genus 2

Assume tha $C$ be the hyperelliptic curve $y^2 = (x-a_1)\cdots (x-a_5)$ of genus $g=2$ and $a_i \in \mathbb{Z}$ and we know that the integers $a_i$ has the form $a_i= d_1^2 - d_i^2$ for some positive ...

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votes

**0**answers

139 views

### Tangent space of Hilbert scheme

We have the following theorem:
Let $X$ be a projective scheme over an algebraically closed field $k$, and $Y \subset X$ a closed subscheme with Hilbert polynomial $P$. Then$$T_{[Y]}\text{Hilb}_P (X) ...

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votes

**1**answer

276 views

### Asymptotic $\int_M \mathrm{exp}[\mathbf{e}\left(n -\frac{t}{2\pi i}\right)] \left( 1 + \frac{5}{6} \mathbf{e}^2 \right)^{1/2} $ on quintic Calabi-Yau

Let $M = \{ G(x) = 0 \} \subseteq \mathbb{P}^4$ be a quintic Calabi-Yau and $\mathbf{e} \in H^2(M, \mathbb{Z})$ such that $\int_M \mathbf{e}^3 = 5$. Then as $t \gg 1$:
$$
\int_M
e^{n \mathbf{e}}
...

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votes

**1**answer

156 views

### Construction of coherent sheaf such that $\text{Proj}\,\text{Sym}\,(\mathcal{F}) = \text{Sym}^n X$

Let $X$ be a smooth projective curve. How do I construct a coherent sheaf $\mathcal{F}$ on $\text{Pic}^n X$ (i.e., the component of the Picard scheme of $X$ parametrizing line bundles of degree $n$) ...

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votes

**0**answers

108 views

### Does the Bombieri-Lang conjecture imply severe restrictions on rational points on twists of hyperelliptic curves?

According to Silverman, the Bombieri-Lang conjecture implies
that the rational points of surface on general type lie on
finite set of curves, except for a finite set of points.
Let $f$ be univariate ...

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votes

**10**answers

4k views

### What is the exact statement of “there are 27 lines on a cubic”?

I think there was a theorem, like
every cubic hypersurface in $\mathbb P^3$ has 27 lines on it.
What is the exact statement and details?

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votes

**0**answers

92 views

### Elementary examples on sheaf extension

Let $V\subset\mathbb{P}^n$ be a projective variety and $C_V$ its conormal subvariety in $T^\ast\mathbb{P}^n$. Denote by $\mathscr{O}_{C_V}$ its structure sheaf, then when will the condition
...

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votes

**0**answers

137 views

### object in D^bCoh(P^2) with prescribed RHom's

Let $\mathbb{C}P^2$ denote the projective plane.
From reading the section of http://homepages.math.uic.edu/~coskun/gokova.pdf
which surveys Gieseker stable sheaves, I have understood that there are ...

**5**

votes

**0**answers

139 views

### Relationship between the syntomic cohomology of Kato and of Fontaine-Messing

Fix a prime $p$ and let $X$ be a $\mathbb{Z}_{p}$-scheme. Write $X_{n}:=X\otimes\mathbb{Z}/p^{n}$ and $\phi:X_{1}\rightarrow X_{1}$ for the absolute Frobenius. Let $X\hookrightarrow Z$ be a (suitable) ...

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votes

**1**answer

202 views

### Ramification divisor on curves in weighted projective space

I was hesitant about posting this question here, but since it deals with a partially unanswered question already on this site I figured that this would be the best place for it. I apologise in advance ...

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votes

**0**answers

127 views

### For all schemes w/Hilbert polynomial P, exists $m_P$ s.t. no higher cohomology, $I(k)$ generated by globally sections, multiplication is surjective

Consider the following theorem.
For every polynomial $P$, there exists an integer $m_P$ such that for all ideal subsheaves $I \subset \mathcal{O}_{\mathbb{P}^n}$ with Hilbert polynomial $P$ and ...

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votes

**0**answers

108 views

### Pull-push formula?

There are many contexts in which the push-pull formula $f_*(f^*(\alpha)\cdot \beta) = \alpha \cdot f_*(\beta)$ holds. I am interesting mostly in the case of algebraic K-theory and Chow rings (under ...

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votes

**1**answer

116 views

### Derived pullback of the coarse moduli morphism

Let $f: \mathcal{X}\to X$ be a morphism from a smooth DM-stack $\mathcal{X}$ to its coarse moduli space $X$. Assume that $X$ is also smooth. Is it true that $Lf^*$ is fully faithful and induces an ...

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votes

**1**answer

231 views

### Definition of p-adic modular forms

I have been reading Hida's book "p-Adic automorphism forms on Shimura varieties" and I don't understand a point.
He first describes p-adic modular forms of tame level N as functions on the Igusa ...

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**0**answers

166 views

### anabelian analogues of the Weil Pairing?

Classically the Weil pairing for an abelian variety $A$ over a field $k$ (say of characteristic 0 for simplicity) can be thought of as an (especially nice) morphism of galois modules:
...

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vote

**1**answer

371 views

### Aysmptotic comparison of L^2 sections versus generating sections

Let $s_1,\ldots, s_k$ be linearly independent global holomorphic sections of a holomorphic line bundle $E$ over a compact algebraic manifold $X$, with volume form $\Omega$.
For $m$ large, let ...

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votes

**3**answers

2k views

### Left invariant metric on SLn

I am looking for a left invariant metric on $SL_n(\mathbb{R})$. If this is not possible, it would be acceptable to have a metric on $SL_n(\mathbb{R})/SO_n(\mathbb{R})$ or something like that. Is there ...

**2**

votes

**1**answer

207 views

### When are direct products exact in the category of quasi-coherent sheaves?

(This question is crossposted from MSE, since there the question did not recieve any attention whatsoever.)
I would like to know if there is a description (or at least some sufficient condition ...

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votes

**3**answers

2k views

### What is Kirillov's method good for?

I am planing to study Kirillov's orbit method. I have seen Kirillov's method in several branch of mathematics, for instance, functional analysis, geometry, .... Why is this theory important for ...

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vote

**0**answers

112 views

### Reference for the statement that the complement of an affine open has codimension one

The following statement seems to be "well-known", but I am unable to find a reference in the standard literature. Could someone suggest a reference?
Let $X$ be a separated normal connected Noetherian ...

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**1**answer

110 views

### The cohomology ring of a compact toric manifold

Given a compact toric manifold $X$ with Picard number $r$ it is well known that the cohomology ring $H^*(X;\mathbb{C})$ is a quotient of the polynomial ring $\mathbb{C}[p_1,\dots,p_r]$ by the ideal ...

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**1**answer

145 views

### Whether the multiplicity changes under a fields extension

Suppose $X$ is a projective scheme (or you can only consider the hypersurface case) in $\mathbb{P}^n$ over an algebraic closed field $k$, and $K/k$ is a field extension. Let $\xi$ be a point (neither ...

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**0**answers

35 views

### Area calculation to determine fit [closed]

How many 27" x 16" rectangles can fit inside of a circle with a 300" diameter without extending past the circle circumference?

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**0**answers

147 views

### Morphism of Shimura varieties and differential equations

Is there a way of constructing a morphism between Shimura varieties using differential equations? Maybe, this looks like a completely ridiculous question, so I think that I should explain the context ...

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vote

**1**answer

317 views

### Soft Question: Relationships Between Moduli Space and Objects They Parametrize

Apologies in advance if this question is not suitable for MO. My friend and I were wondering recently what, if any, are the relationships between the geometric properties of a moduli space and the ...

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votes

**1**answer

110 views

### Definition of Strongly Stable 0-cycle

I am not sure whether this question deserves to be asked in this forum, but I have no other choice as I can't find the definition anywhere. So here is the question:
When is a 0-cycle on $\mathbb P^n$ ...

**2**

votes

**0**answers

77 views

### Combinatorial interpretation for a toric intersection number

Let $X$ be an $n$-dimensional toric variety and let $D$ be an effective divisor (eg nef or ample). Is there a combinatorial interpretation (eg in terms of the fan or polytope) of the intersection ...

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votes

**1**answer

265 views

### Serre duality over a non-algebraically closed field

Suppose $X$ is a projective smooth variety over a non-algebraically closed field , do we still have $Ext^i(F,\omega)\to H^{n-i}(X,F)^{\vee}$? (Hartshorne's proof Thm III 7.6 requires $k$ to be ...

**2**

votes

**0**answers

140 views

### Rank of the Jacobian of twists of hyperelliptic curves

Suppose that a hyperelliptic curve $C$ of genus $g \geq 4$ is given by the equation
$$\displaystyle C: y^2 = a_0 x^{2g+2} + a_1 x^{2g+1} + \cdots + a_{2g+2} = f(x).$$
The Jacobian variety $J(C)$ of ...

**1**

vote

**0**answers

62 views

### Is the elementary transformation along a curve decomposable?

Let $S$ be a surface. Let $L$ be an ample line bundle on $S$. Let $C\in |L|$ be a curve on $S$, and let $A$ be a globally generated line bundle on $C$ of degree $d$ and with 2 sections.
Then we get ...

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votes

**1**answer

113 views

### perfect modules over polynomial algebra

This may be obvious. My question is short:
$R$ is the polynomial algebra $\mathbb{k}[X_{1},\dots , X_{n}]$. Is the $R$-module $\mathbb{k}$ perfect in the sense that $\mathbb{k}$ is a compact object ...

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votes

**1**answer

268 views

### irreducibility of general fiber

I would like to get a reference of the following fact.
Let $A\subseteq B$ be affine domains over an algebraically closed field of characteristic zero. If $Q(A)$ is algebraically closed in $Q(B)$, ...

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votes

**2**answers

685 views

### Varieties with an ample vector bundle mapping to their tangent bundle

A well-known result of Andreatta and Wisniewski says: Let $X$ be a projective complex manifold whose tangent bundle $T_X$ contains an ample sub-bundle $\mathscr{E}$. Then $X$ is isomorphic to ...

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votes

**3**answers

201 views

### Example of indecomposable self injective ring

Is there any example of an indecomposable self-injective commutative ring with 4 or more maximal ideals?$$$$$$$$

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**1**answer

243 views

### Castelnuovo-Mumford Regularity of Ideals of Maximal Minors

Hi,
I have a $m \times 2m$ matrix of linear forms over $\mathbb{C}[x,y,z,w]$. It is of the form $M = ( x I - A z -B w | y I - C z - D w$. Here $A,B,C$ and $D$ are $m \times m$ scalar matrices. Let ...

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**0**answers

338 views

### Easiest way to see Theorem 1.2 of Deligne-Mumford's “The irreducibility of the space of curves of given genus”?

Theorem 1.2 of Deligne-Mumford's 1969 IHÉS paper, "The irreducibility of the space of curves of given genus," is as follows.
If $g \ge 2$ and $C$ is a stable curve of genus $g$ over an ...

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vote

**0**answers

226 views

### Generic triviality of $G$-bundles

Let $k$ be an algebraically closed field and $X$ a curve over $k$. Then any $G$-bundle on a curve (where $G$ is reductive and connected) is generically trivial. This is the one of the main results of ...

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votes

**1**answer

182 views

### Hyperelliptic curves with fixed genus and many rational points

It is a famous theorem of Faltings, previously a conjecture by Mordell, that any algebraic curve of genus at least $2$ defined over the rational numbers have at most finitely many rational points. A ...

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**0**answers

45 views

### Hermitian Matrices over Quaternions with Rank at most k [migrated]

The set of Hermitian matrices of the form: $X+iY+jW+kZ$ with $X,Y,Z,W \in \mathbb{C}^{M x M}$. $X$ symmetric, and $Y,Z,W$ skew-symmetric, with $rank(X+iY+jW+kZ)\leq{k}$, has what dimension as a ...

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**1**answer

140 views

### dimension of singular locus and complete intersection of a hypersurface

Let $X$ be a reduced projective hypersurface over a field $k$, which is defined by the homogeneous equation $f(T_0,\ldots,T_n)=0$. If the dimension of the singular locus of $X$ is $s$, $0\leq s\leq ...

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votes

**1**answer

283 views

### Is there a geometric proof for the upper semicontinuity of fiber dimension in algebraic geometry?

One of the first theorems encountered in algebraic geometry is the upper semicontinuity of fiber dimension:
Let $ f : X \to Y $ be a surjective regular map between irreducible varieties with ...

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votes

**2**answers

656 views

### Incidence geometry and matrices

Supposing I have a $0/1$ or $\pm1$ matrix $A$ of size $m\times n$, is there a minimum $d$ (that works for every $m\times n$ $A$) such that there exists $m$ lines $r_1,\dots,r_m$, $n$ lines ...