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

**4**

votes

**1**answer

468 views

### Why is the inverse of a bijective rational map rational?

Let $f:X\to Y$ be a bijective rational map of an open dense subset $X$ of $\mathbb{C}\times\mathbb{C}$ onto an open dense subset $Y$ of $\mathbb{C}\times\mathbb{C}$. How to prove that the inverse map ...

**2**

votes

**0**answers

118 views

### Varieties with Chow groups supported in positive codimension: examples and properties?

In their 1983 paper "Remarks on correspondences and algebraic cycles" Bloch and Srinivas proved several interesting properties of smooth proper varieties (over universal domains) whose Chow groups of ...

**1**

vote

**0**answers

93 views

### G-invariant vector bundle over schemes?

Take X to be a scheme over a (say algebraic closed) field k and G a group scheme act on X. I want to get more convenient knowledge about G-invariant vector bundles over X.
I'm inspired by this ...

**4**

votes

**1**answer

166 views

### $n$ columns of a specific “infinite” Vandermonde matrix always linearly independent? [closed]

Consider the "infinite" Vandermonde matrix
$$
V (x_1, x_2, \ldots , x_n) =
\begin{pmatrix}
1 & x_1 & x_1^2 & \cdots & x_1^{n-1} & x_1^n & x_1^{n+1} & \cdots \\
...

**1**

vote

**0**answers

104 views

### Why $D^b(S)\cong D^b_{\text{Car}}(\text{cosq}(X\rightarrow S))$?

Let $p: X\rightarrow S$ be a map between topological spaces and we can construct the simplicial space $\text{cosq}(X\rightarrow S)$ where $X_0=X$, $X_1=X\times_S X$ and
$$
X_n=\underbrace{ X\times_S ...

**2**

votes

**1**answer

206 views

### On a property of the Grothendieck group of a smooth projective curve

Let $K$ be a complete DVR of characteristic $0$, $X$ a smooth projective curve over $K$. Denote by $K^0(X)$ the Grothendieck group of locally free sheaves on $X$ and by $\mbox{det}$ the natural group ...

**0**

votes

**0**answers

109 views

### Descent datum for a line bundle

Let $\pi:C \to \mathbb P^1$ be a double cover branched at $r$ points. To understand the theory of descent better, I would like, if possible, to construct by hands the descent datum of a line bundle ...

**15**

votes

**3**answers

1k views

### Classification of rings satisfying $a^4=a$

We have the famous classification of rings satisfying $a^2=a$ (for each element $a$) in terms of Stone spaces, via $X \mapsto C(X,\mathbb{F}_2)$. Similarly, rings satisfying $a^3=a$ are classified by ...

**2**

votes

**0**answers

140 views

### l-adic cohomology and perverse sheaves

Let consider the map $tr:\mathbb{G}_{m}^{n}\rightarrow\mathbb{A}^{1}_{\mathbb{F}_{q}}$ given by the sum of the coordinates and let $\psi:\mathbb{F}_{q}\rightarrow\mathbb{Q}_{l}^{*}$ a non trivial ...

**1**

vote

**0**answers

85 views

### Intuition for the structure theorem for connected solvable algebraic groups over an algebraically closed field of positive characteristic

In the theory of algebraic groups one of the fundamental results is the structure theorem for connected solvable groups. I never understood the proof in any of the standard textbooks, so I just moved ...

**3**

votes

**1**answer

176 views

### Spec of an injective ring map contains minimal primes in its image?

Let $f\colon A \rightarrow B$ be an injective ring homomorphism. One knows (from EGA I, 1.2.7 or elsewhere) that the image of $\mathrm{Spec}(f)$ is dense. Does that image necessarily contain all the ...

**2**

votes

**0**answers

80 views

### question on n-th root coverings

Let $X$ be a smooth projective complex variety and $D \subset X$ an effective divisor. Assume that there is a line bundle $L$ such that $L^n=\mathcal{O}(D)$ for some integer $d \geq 2$.
Suppose $s ...

**0**

votes

**0**answers

182 views

### “Spreading out” locally free sheaves

Let $Y$ be a regular surface flat, projective over $R$, where $R$ is complete DVR. Let $X$ be the generic fiber and $F$ be a locally free sheaf on $X$. We know that if the rank of $F$ is $1$ then we ...

**0**

votes

**0**answers

61 views

### birational equivalence of linear sections of algebraic varieties

Let $X$ be an irreducible algebraic variety and suppose that $L$ is a linear space defined by the linear forms $l_1,l_2,\ldots,l_k$. I want to study $L\cap X$. I would like to know whether the ...

**3**

votes

**0**answers

136 views

### Cancellation and splitting theorems for vector bundles etc over schemes

It is not too hard, in the theory of vector bundles over manifolds (or nice topological spaces, say locally contractible with finite covering dimension), to arrive at a splitting theorem. This ...

**0**

votes

**0**answers

109 views

### A question on a toric singularity

Recently I was told that the threefold singularity
$$
x^2+y^2+z^2+w^{2n}=0
$$
in $\mathbb{C}^4$ is a toric variety. How can I see this? What are the generators of the toric fan? Moreover, is it true ...

**1**

vote

**1**answer

151 views

### pencil of quadrics consisting of singular quadrics

A pencil $l$ of quadrics in $\mathbb{P}^4_{<x_0,\cdots,x_4>}$ consists of singular quadrics only if:
(1) quadrics in $l$ have a common singular point; or
(2) quadrics in $l$ contain a common ...

**0**

votes

**0**answers

174 views

### Holomorphic vector bundles over $\mathbb{CP}^1$ and elliptic curves

Can anyone suggest a good exposition of the classification of holomorphic vector bundles over $\mathbb{CP}^1$? Also does there exist any analytic or more elementary proof of Atiyah's classification of ...

**10**

votes

**1**answer

588 views

### Is there an analog of the Birch/Swinnerton-Dyer conjecture for abelian varieties in higher dimensions?

I am wondering if there is a multi-dimensional analog of the
Birch/Swinnerton-Dyer (BSD) conjecture.
The recent famous result inching toward resolution of that conjecture is:
Bhargava, Manjul, and ...

**1**

vote

**0**answers

74 views

### Can we classify reductive group schemes over curves

Let $C$ be a smooth quasi-projective connected curve over the complex numbers.
Can one classify all reductive group schemes over $C$?
Certainly, you have the trivial ones (coming from pulling-back ...

**4**

votes

**1**answer

112 views

### Canonical bundle of moduli space of rational curves and automorphisms

Let $\overline{M}_{0,n}$ be the usual Deligne-Mumford compactification of $M_{0,n}$ the moduli space of smooth $n$-pointed rational curves.
The canonical divisor $K_{\overline{M}_{0,n}}$ can be ...

**2**

votes

**0**answers

62 views

### Criterion for the existence of finite locally free resolution

Let $X$ be a projective variety over an algebraically closed field $k$, $S$ be a $k$-scheme, $E$ be a coherent sheaf on $X \times_k S$, flat over $S$. We know that if $X$ is smooth then $E$ has a ...

**1**

vote

**0**answers

132 views

### Is there a D-module theory in characteristic p>0

everyone.
In characteristic 0 we have a good theory of D-modules. In particular, we have a formalism of Grothendieck's six operators in the derived category of holonomic D-modules and Riemann-Hilbert ...

**2**

votes

**0**answers

103 views

### subschemes of abelian scheme over artinian basis

Let $R$ be an artinian thickening of a field $k$. Denote with $S=Spec(R)$. Let $A$ be an abelian scheme over $S$. Let $X$ be a closed, reduced, equidimensional subscheme of the special fiber $A_k$. I ...

**2**

votes

**1**answer

121 views

### convergence of L-functions of curves

Let $C$ be a smooth projective curve over $\mathbb{Q}$. Its associated L-function is defined by
$$
L(C, s)=\prod_{p \text{ prime}} L_p(C, s),
$$
where, if $p$ is a prime of good reduction, ...

**7**

votes

**0**answers

130 views

### Vector bundles with flat connections

Let $V$ and $W$ be varieties defined over a number field, and let $f: V \rightarrow W$ be a map. Suppose that the induced map on the complex points of $V$ and $W$ induces an isomorphism on fundamental ...

**2**

votes

**0**answers

102 views

### Questions about transformation or integral transformation

I have asked several mathematicians about the following questions,but all of them think they are good questions,but can not give a complete answer.Now I have to come here to ask mathematicians all ...

**4**

votes

**0**answers

71 views

### Looking for some abelian surface fibration

Do you know of any explicit smooth complex projective threefold $X$ with an Abelian surface fibration over $\mathbb{P}^1$ such that $K_X = [-F]$ where $F$ is the fiber class divisor.
I am not ...

**3**

votes

**1**answer

148 views

### Subspaces of Grassmannian under Plucker embedding

The Grassmannian $G(k,V)$ is the set of all $k$-dimensional subspaces of a vector space of dimension $n$. It can be embedded inside $\mathbb{P}(\bigwedge^k V)\cong \mathbb{P}^{\binom{n}{k}-1}$ as a ...

**4**

votes

**0**answers

86 views

### Strategy to prove formula for top chern class from knowlege of chern character

I am trying to prove a conjecture that involves an enumerative problem. In the course of doing so, the following situation came up.
I have a sequence of (smooth, complex, rationally connected) ...

**8**

votes

**2**answers

322 views

### “Forms” of quadrics

The theory of Severi-Brauer varieties is well-known. Let $k$ be a (perfect) field. There may exist varieties not isomorphic to $\mathbf{P}^n$ over $k$, which are isomorphic to $\mathbf{P}^n$ over ...

**6**

votes

**1**answer

202 views

### Is there an $(\infty,2)$-category with morphisms given by $D^b\text{Coh}$?

My question is:
Has anyone constructed an $(\infty,2)$-category whose objects are (projective, maybe smooth, ...) varieties, and where the 1-morphisms from $X$ to $Y$ are given by ...

**2**

votes

**1**answer

140 views

### The stable general linear group in algebraic geometry

Let $A$ be a $ \mathbb{C}$-algebra. Then we have the following increasing union:
$$ {\rm GL}_1(A) \subseteq {\rm GL}_2(A) \subseteq {\rm GL}_3(A) \subseteq \dots \subset {\rm GL}(A) $$
We call $ {\rm ...

**4**

votes

**2**answers

213 views

### Non trivial family of hyperelliptic curves

Let $X$ ba a smooth hyperelliptic curve of genus $g$, and let $f:X\rightarrow X$ be the hyperelliptic involution. Consider a $K3$ surface $S$ with an involution $g$ without fixed points. The quotient ...

**3**

votes

**0**answers

78 views

### birational classification of rationally connected 3-folds

What is the birational classification of (smooth projective) rationally connected 3-folds (over algebraically closed fields of characteristic $0$ or even $\mathbf{C}$, if $\mathrm{char}(k) = p > ...

**2**

votes

**1**answer

153 views

### discriminant of smooth quartic del Pezzo surface in $\mathbb{P}^4$

I can't understand the proof of Lemma3.3 in Stability of genus 5 canonical curves.
Let $C$ be a complete intersection of three quadrics in $\mathbb{P}^4$ and let $\Lambda$ be the net of quadrics ...

**1**

vote

**0**answers

201 views

### Two questions on canonical line bundle over $\mathbb{C}P^{n}$

The canonical line bundle over $\mathbb{C}P^{n}$ is denoted by $\ell_{n}$. It is well known that: $$(\ell_{1}\otimes \ell_{1})\oplus1 \simeq\ell_{1}\oplus\ell_{1}$$ Both sides are isomorphic ...

**2**

votes

**1**answer

239 views

### Stalks of étale sheaves

I want to prove that $0 \to F\to G\to H \to 0$ is an exact sequence of étale sheaves. I understand that it is enough to show that $0\to F_{\bar{x}}\to G_{\bar{x}}\to H_{\bar{x}}\to 0$ is exact at ...

**1**

vote

**1**answer

134 views

### L-function of twist

I'd like to ask the following easy question, since I can't find a reference.
Let $X/\mathbb{Q}$ a smooth projective variety. How does one express the $L$-function of the twist, $L(H^i(X)(r), s)$ in ...

**10**

votes

**1**answer

561 views

### Ways to show a system of polynomial equations has no solution

I came across the following system of polynomial equations on $X_1,\dots,X_{m-2}$:
$$
\begin{cases}
2X_{2s}+\sum\limits_{t=1}^{2s-1}(-1)^tX_tX_{2s-t}=0,\quad s=1,\dots,\frac{m}{2}-1,\\
...

**1**

vote

**0**answers

212 views

### Learning roadmap in Algebra [closed]

I am a senior undergraduate student in mathematics, I have a sound knowledge in the following areas:
a) Commutative Algebra
b) Field Theory and Galois Theory
c) Homological Algebra
My question is ...

**2**

votes

**1**answer

82 views

### u-Invariants of p-adic function fields

In his Paper "Fields of u-invariant 9" Oleg Izhboldin points out that for a algebraic closed, finitely generated field $k$ we have $u(k)= 2^{cd(k)}$. In particular we have
...

**3**

votes

**0**answers

107 views

### Singularities in mixed characteristic

Let $R$ be a regular local ring in mixed characteristic. Moreover, I assume that $R$ is the local ring of a point on a smooth $\mathbb Z_p$-scheme and that $R/pR$ is regular. ($\mathbb Z_p$ is the ...

**3**

votes

**1**answer

166 views

### Compositional inversion and generating functions in algebraic geometry

The exponential generating function of the graded dimension of the cohomology ring of the moduli space of n-pointed curves of genus zero satisfying the associativity equations of physics (the WDVV ...

**2**

votes

**0**answers

60 views

### Algebraic approach to showing trigonometric equations have no solution

I have very little background in algebra and algebraic geometry, so please bear with me.
I am trying to show that certain systems of trigonometric polynomial equations generally have no solution. One ...

**5**

votes

**1**answer

129 views

### p-adic L-function of curves

Given a smooth projective curve $C$ over $\mathbb{Q}$ one has the $L$-function $L(C, s)$ and the Beilinson conjectures predict its values at integers $s=n$ in terms of regulators.
Is there a p-adic ...

**3**

votes

**2**answers

194 views

### Sheaf of isogenies representable?

It is well-known that "the" stack of elliptic curves (allow me to be vague as to singular curves, compactifications etc) has a presentation by a groupoid in schemes. One of the things that needs to be ...

**1**

vote

**1**answer

99 views

### Compute higher direct image for Gm under open embedding

Let $U \subset \mathbb P^1$ be an open subset of projective line (over $\mathbb C$) after removing $r$ points and $j: U\hookrightarrow \mathbb P^1$ an open immersion. How do I compute $R^1j_*\mathbb ...

**4**

votes

**1**answer

185 views

### transcendence of beta values

(1) Can anybody suggest a readable reference for Schneider's theorem that the number
$$
\beta(a, b)=\frac{\Gamma(a)\Gamma(b)}{\Gamma(a+b)}
$$ is transcendental for $a, b \in \mathbb{Q}$ such that none ...

**1**

vote

**0**answers

73 views

### plane curves with two points of high multiplicity

Let $\mathcal{C}$ be an irreducible plane curve in $\mathbb{P}^2_\mathbb{C}$ of degree $d$. Let $D$ be a quartic with three irreducible components with normal crossing singularities, i.e. a conic and ...