# Tagged Questions

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

**6**

votes

**1**answer

121 views

### For a Sum-of-Squares cost functions J(x) is it true that J(x)-j* is also SOS?

For polynomial optimization problems the sum-of-squares theory and Lasserre relaxation hierarchy provides a theoretically handy way of getting the solution. There are also results saying that finite ...

**11**

votes

**2**answers

525 views

### The quotient stack $[\mathbb{A}^n / \mathrm{GL}_n]$

Consider the affine space $\mathbb{A}^n$ (over some base scheme) with the usual $\mathrm{GL}_n$-action. What does the quotient stack $[\mathbb{A}^n / \mathrm{GL}_n]$ classify? If $n=1$, then we get ...

**2**

votes

**2**answers

249 views

### Compact elements in $G(K)$ for a reductive group $G$ over a nonarchimedean local field $K$

Let $K$ be a nonarchimedean local field and $G$ a (connected) reductive group over $K$, so that $G(K)$ carries a natural topology. An element $g \in G(K)$ is compact if it is contained in a compact ...

**0**

votes

**1**answer

112 views

### Is pushforward of an ample divisor under small birational map nef?

Let $X, Y$ be $\mathbb{Q}$-factorial, projective, normal varieties. Let $f: X --> Y$ be a small birational map. I have two related questions about pushforward of an ample divisor:
(1) Let $H_X$ be ...

**3**

votes

**2**answers

515 views

### Two basic questions on derived categories

Let $\mathcal{A}, \mathcal{B}$ be two abelian categories with sufficiently many injective objects (in my case these are categories of sheaves of vector spaces on a manifold).
Let $f_*\colon ...

**8**

votes

**1**answer

436 views

### Are there noncommutative extensions of $\alpha_p$ by $\mathbb{G}_m$?

Let $k$ be a field of characteristic $p > 0$ (algebraically closed, if you want; that doesn't make a difference). Consider a finite type $k$-group scheme $E$ that is a (central) extension of ...

**0**

votes

**0**answers

67 views

### on lifting elements in a tangent space

Let X a normal integral scheme over a base field scheme, assumedd to be singular and an integer $n$
Let $\mathcal{O}=k[[t]]$, we consider the arc space $X(\mathcal{O})$ which is a $k$- pro-scheme and ...

**0**

votes

**1**answer

78 views

### Vector bundle on ruled surface $X\times \mathbb{P}^{1}$

let $E$ be a vector bundle of rank $r$ on $X\times \mathbb{P}^{1}$ where $X$ is a smooth projective curve. Assume now that $E|_{F_{p}} \cong \mathcal{O}_{\mathbb{P}^{1}}^{r}$ for any p-fiber where $p: ...

**1**

vote

**1**answer

116 views

### Some questions about ruled surfaces defined over $\overline{\mathbb Q}$

definitions:
A non-singular complex projective surface $S$ is a ruled surface if it is birationally equivalent to $C\times_{\text{Spec} \mathbb C}\mathbb P^1_{\mathbb C}$ where $C$ is a non-singular ...

**0**

votes

**0**answers

78 views

### Sufficient conditions to get complete intersection curves

Let $H_1,H_2\cdots,H_{d-1}$ be hypersurfaces in $\mathbb{P}^d$, if the intersection $B:=H_1\cap H_2\cap \cdots \cap H_{d-1}$ is $1$-dimensional then it is called a complete intersection curve.
What ...

**0**

votes

**1**answer

196 views

### Line on a hyper surface

Assume $X$ is a hyper surface in $\mathbb{P}^n$, can one always find a closed immersion $i:\mathbb{P}^1 \rightarrow X$?

**2**

votes

**1**answer

173 views

### Number of minimal models of a surface

I would like to know if the following statement is true or false:
Given a non-singular complex projective surface $S$, it has at most a countable number of minimal models (up to isomorphism).
...

**2**

votes

**1**answer

243 views

### When is the Hodge diamond concentrated in $H^{n,n}$'s?

Let $X$ be a smooth projective complex algebraic variety. The Hodge decomposition tells us that $H^n(X, \mathbf C) = \oplus H^{p,q}$.
Here is my question:
For what kind of $X$ is $H^{2n}(X) = ...

**0**

votes

**2**answers

134 views

### Decompose a big divisor as nef big divisor and effective divisor

Let $W_n$ be a set of a log pair having the following property:
For any $(X, D) \in W_n$
(1)$X$ has dimensional $n$ with tirvial canonical divisor (i.e.$K_X = 0$). Moreover, $X$ is a ...

**1**

vote

**2**answers

222 views

### When the contraction is a morphism defined over $\overline{\mathbb Q}$

Suppose that $S$ is a complex projective surface defined over $\overline{\mathbb Q}$, namely there exists a surface $S_{\overline{\mathbb Q}}$ over $\overline{\mathbb Q}$ such that:
...

**2**

votes

**1**answer

154 views

### About $\mathbb P^1_\mathbb C$ contained in a surface

Suppose that $X$ is a non-singular projective surface over $\mathbb {\overline Q}$ ( $X$ is a $\mathbb {\overline Q}$-scheme...) and suppose that there is an embedding:
$$j:\mathbb P^1_{\mathbb ...

**5**

votes

**0**answers

150 views

### Lower semicontinuity of naive fiber size

I would like to present the following result in my algebraic geometry class, but it is seeming much harder than I would expect. Since my class is working with closed points over an algebraically ...

**2**

votes

**0**answers

100 views

### Do all full exceptional sequences of a triangulated category have the same length?

Let $\mathcal{D}$ be a $k$-linear triangulated category and $\langle E_n,E_{n-1},\ldots, E_0\rangle$ be a sequence of objects of $\mathcal{D}$. We call $\langle E_n,E_{n-1},\ldots, E_0\rangle$ an ...

**0**

votes

**1**answer

258 views

### Is there a Gröbner basis analogue that exists for vector spaces?

Suppose I have a coordinate system $t_1,\ldots t_N$ with a lexicographical ordering. Let LT denote choosing the lowest term of a polynomial with respect to this ordering. e.g. LT$(t_1 + t_2)=t_2$.
...

**0**

votes

**0**answers

86 views

### Reciprocity laws in different dimensions

Let $M/L/Qp$ be a finite galois abelian extension of local fields and define
$\mathcal{M}=M\{\{T\}\}=\{\sum_{i\in \mathbb{Z}}a_iT^i:a_i\in M,\min_{i\in \mathbb{Z}}, v(a_i)>−\infty , \lim_{i\to ...

**1**

vote

**0**answers

64 views

### Relation between 1-dimensional and 2-dimensional reciprocity maps

Let $M/L/\mathbb{Q}_p$ be a finite galois abelian extension of local fields and define
$\mathcal{M}=M\{\{T\}\}=\{\sum_{i\in \mathcal{Z}}a_iT^i : a_i\in M, \min_{i\in \mathcal{Z}} v(a_i)>-\infty, ...

**4**

votes

**0**answers

109 views

### On the cohomology ring of the Hilbert scheme of points on k3 or abelian surfaces

There are many results on the cohomology of the Hilbert scheme of points of a surface.
Gottsche calcaluted the Betti numbers and Nakajima got the generators of the cohomology. Also
there are results ...

**0**

votes

**1**answer

209 views

### Improving Newton's Inequalities using the Taylor Theorem

Newton's inequalities say that if $f(x) = \sum \binom{n}{k} a_k x^k$ is a polynomial with all real roots then $ a_k^2 > a_{k-1}a_{k+1}$.
The proof this result uses that if $f(x)$ has all real ...

**2**

votes

**1**answer

187 views

### Minimal model of a non-singular complex projective surface defined over $\overline{\mathbb Q}$

Suppose that $S$ is a non-singular complex projective surface that is defined over $\overline{\mathbb Q}$, namely $S\cong\text{Proj}\frac{\mathbb C[T_1,T_2,\ldots,T_n]}{(f_1,\ldots,f_n)}$ where $f_i$ ...

**10**

votes

**4**answers

707 views

### Soft question: beginners reference to moduli spaces

What is a geometrically intuitive yet reasonably general first introduction to the theory of Moduli spaces?
(Possibly introducing stacks also)?
I'm looking for something which really gets the pictures ...

**1**

vote

**1**answer

232 views

### Does every hypersurface in the projective plane contain a projective line?

Consider $P^{2}(\mathbb{C})$, the space of all lines through the origin in $\mathbb{C}^{3}$ (or $\mathbb{R}^3$ if that works better). Let $X\subset P^{2}(\mathbb{C})$ be a (nonempty) hypersurface ...

**3**

votes

**1**answer

82 views

### simplex in hyperbolic space and quadrics in projective space

I am trying to understand Goncharov's paper "Volumes of hyperbolic manifolds and mixed Tate motives", http://www.ams.org/journals/jams/1999-12-02/S0894-0347-99-00293-3/S0894-0347-99-00293-3.pdf
In ...

**9**

votes

**0**answers

189 views

### How tight is the Weil bound for this exponential sum?

Let $f$ be a nonconstant polynomial of degree $d$. Let $\psi$ be a character of the additive group of $\mathbb F_p$. By Weil, we have:
$$ \left| \sum_{x\in \mathbb F_p} \psi ( f(x)) \right| \leq ...

**0**

votes

**0**answers

92 views

### What can one say about zero-cycle groups for products of Chow motives

What can one say about the Chow group of zero-cycles (up to rational equivalence) for a product of smooth projective varieties and Chow motives (so, I am interested in the kernel $Chow_0(P)\otimes ...

**0**

votes

**0**answers

82 views

### G-equivariant coherent sheaves on Bott-Samelson Resolutions

Let $G$ be a Lie group, $B$ be a Borel subgroup. $G/B$ is the corresponding flag variety.
Let $w$ be an element of the Weyl group $W$ with a reduced expression
$w = s_1 \cdots s_n$. Let $X_w$ be ...

**5**

votes

**1**answer

162 views

### Intermediate Jacobians of intersections of two quadrics

Let
$$X: \quad Q_1(x)=Q_2(x) = 0 \quad \subset \mathbb{P}^{2n+1},$$
be a smooth complete intersection of two quadrics of odd dimension over a field $k$, not of characteristic $2$. Let $J(X)$ denote ...

**0**

votes

**0**answers

52 views

### Toroidal embeddings over a DVR

I'm starting to study toroidal embeddings over a Discrete Valuation Ring $R$.
In particular i refer to Mumford's section in the book "Toroidal embeddings I".
Let $I=(\pi)$ be the maximal ideal, ...

**1**

vote

**1**answer

95 views

### Zariski closure of the boundary of a closed convex subset of ${\mathbb R}^n$

Let $\eta$ be a closed convex subset of ${\mathbb R}^n$ of convex dimension $n$ (not necessarily compact) with nonempty boundary $\partial \eta$. Then $\eta$ is an $n$-dimensional topological ...

**1**

vote

**1**answer

144 views

### How to understand a rooting of a dessin d'enfant?

As I understand it, rooted maps on surfaces were first introduced in enumerative combinatorics because they are easier to count than unrooted maps, which can have non-trivial symmetries. A map is a ...

**1**

vote

**0**answers

111 views

### Thickness of the category of perfect complexes with finite length homology

Let $R$ be a commutative Noetherian local ring and let $D(R)$ be the derived category of $R$-modules. Recall that a chain complex $C_\bullet$ of modules over $R$ is called perfect if it is isomorphic ...

**0**

votes

**0**answers

78 views

### How can information about morphisms between modular curves be read from the morphism of their corresponding stacks?

$\newcommand{\yY}{\mathcal{Y}}$
$\newcommand{\PSL}{\text{PSL}}$
$\newcommand{\SL}{\text{SL}}$
$\newcommand{\ZZ}{\mathbb{Z}}$
$\newcommand{\CC}{\mathbb{C}}$
$\newcommand{\mM}{\mathcal{M}}$
Let $Y(1)$ ...

**0**

votes

**1**answer

113 views

### Is the cokernel of a map of sheaves a seperated presheaf?

The cokernel of a map of sheaves is not necessarily a sheaf until you sheafify. In every example I have seen of the cokernel failing to be a sheaf it is the glueability axiom that fails while the ...

**2**

votes

**1**answer

142 views

### On the generating series of degree $d>1$ Gromov-Witten invariants of the local $\mathbb P^1$

Let $N$ be the total space of the vector bundle $\mathscr O_{\mathbb P^1}(-1)\oplus \mathscr O_{\mathbb P^1}(-1)$ over $\mathbb P^1$, and let $C_0\subset N$ be the zero section. Then $N$ is a ...

**6**

votes

**1**answer

253 views

### Can we normalize a complex analytic space in a covering of an open subset?

Let $X$ be a normal connected complex analytic space, $x\in X$ a point, $f$ a nonzero holomorphic function vanishing at $x$. Denote by $U\subseteq X$ the locus where $f$ is nonzero. Suppose that ...

**1**

vote

**0**answers

59 views

### Uniform bounds of number of integral points on affine varieties

In Duke-Rudnick-Sarnak 93, Density of integer points on affine homogeneous varieties, one of the consequences is the following,
Consider the variety $V_{n,k} = \{A \in Mat_n(\mathbb{Z}): det(A) = ...

**3**

votes

**1**answer

175 views

### Flatness of Normalization of regular schemes

I have a followup to the following question: Flatness of normalization.
Suppose that $X$ is a regular scheme (of finite type over a $\mathbb{C}$ if one wants) and $X'$ is the normalization of $X$ in ...

**1**

vote

**3**answers

299 views

### smooth connected affine scheme over Z has good reduction almost everywhere

Let $f(x_1,\ldots,x_n)\in\mathbf{Z}[x_1,\ldots,x_n]$ be a polynomial. Assume that the variety cut out by $f$ is smooth and connected (so irreducible) over $\overline{\mathbf{Q}}$. Where can I find a ...

**2**

votes

**1**answer

104 views

### Rational mapping related to cubic surfaces

A theorem in Manin's book on cubic forms leads to the conclusion that the least degree of a rational mapping from $\mathbb{P}^2$ to general cubic surfaces(e.g. Picard rank=1) over $\mathbb{Q}$ is 6.
...

**1**

vote

**0**answers

131 views

### Which “concrete” morphisms of varieties and motives induce bijections of their lower Chow groups?

This question is a continuation of Varieties with Chow groups supported in positive codimension: examples and properties?
What examples are known of morphisms of varieties and Chow motives (say, over ...

**0**

votes

**0**answers

146 views

### Surjectivity of the algebraic K-functor

Let $R \to S$ be a surjective morphism of commutative rings. For a fixed integer $q$, is there any known condition under which the resulting morphism of the K-groups, $K_q(R) \to K_q(S)$ is ...

**5**

votes

**1**answer

223 views

### Analogy between connections and $\ell$-adic sheaves: what happens with the residue?

There are many analogies between $\ell$-adic sheaves on varieties over finite fields and vector bundles with connections on varieties over fields of characteristic zero. I would like to know what is ...

**0**

votes

**0**answers

70 views

### Vanishing of the module of differentials of a extension of perfect fields

Let $L|F$ be a extension of perfect fields of characteristic $p$, $\phi_F:F \to F_{\phi}$, $\phi_L:L \to L_{\phi}$ the Frobenius isomorphisms ($F_{\phi}=F$ but considered as $F$-algebra via $\phi_F$). ...

**5**

votes

**1**answer

138 views

### Modular polynomials for elliptic curves point counting

The Schoof-Elkies-Atkin (SEA) algorithm (for counting points on elliptic curves over a finite field) performs computations over polynomials modulo some modular polynomials. Originally the "classical" ...

**5**

votes

**1**answer

203 views

### Connected components of the complement of a degree-d affine hypersurface

Let $n$ and $d$ be positive integers, and $f\in\mathbb{R}[x_1,\dots,x_n]$ be a polynomial of degree $d$. Let's consider the zero-set $M = \{x \in \mathbb{R}^n: f(x) = 0\}$ of $f$.
Can we estimate ...

**6**

votes

**1**answer

787 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 ...