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

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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 ...
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1answer
75 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: ...
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1answer
106 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 ...
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73 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 ...
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1answer
192 views

Line on a hyper surface

Assume $X$ is a hyper surface in $\mathbb{P}^n$, can one always find a closed immersion $i:\mathbb{P}^1 \rightarrow X$?
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1answer
166 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
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1answer
225 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) = ...
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2answers
128 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 ...
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2answers
215 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
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1answer
151 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 ...
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141 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 ...
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95 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 ...
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1answer
250 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$(t_1 + t_2)=t_2$. ...
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84 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 ...
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0answers
58 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, ...
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93 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 ...
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1answer
149 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
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1answer
180 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 $f_i$ ...
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4answers
615 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 ...
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1answer
204 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 ...
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1answer
487 views

An Intriguing Tapestry: Number triangles, polytopes, Grassmannians, and scattering amplitudes

$$Prelude \;\;on \;\;Two\;\; Threads$$ Last month the workshop New Geometric Structures in Scatering Amplitudes was hosted by CMI with the following partial overview: Recently, remarkable ...
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1answer
79 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 ...
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174 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 ...
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82 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 ...
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74 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 ...
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1answer
153 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 ...
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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, ...
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1answer
93 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 ...
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1answer
138 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 ...
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0answers
104 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 ...
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72 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 $Y(1)$ ...
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1answer
107 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
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1answer
131 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 ...
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75 views

Comparing two definitions of determinant of coherent sheaves [migrated]

Let $f:X \to S$ be a smooth, projective morphism of $k$-schemes for some field $k$. Let $\mathcal{F}$ be a coherent sheaf on $X$ flat over $S$. We know (by Proposition $2.1.10$ of Huybrechts-Lehn, ...
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1answer
243 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 ...
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55 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
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1answer
120 views

Flatness of Normaliztion of regular schemes

I have a followup to the followig 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 a ...
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3answers
294 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
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1answer
98 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. ...
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128 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 ...
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141 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 ...
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1answer
207 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 ...
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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 $\phi_F$). ...
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138 views

Existence of a rational function on a nonsingular curve with a simple zero at P and order 0 at Q [migrated]

Let $X$ be a nonsingular curve over an algebraically closed field $k$ (by curve over $k$, I mean an integral separated scheme of finite type over $k$ which has dimension 1). Let $P$ and $Q$ be ...
5
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1answer
121 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" ...
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1answer
191 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 ...
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1answer
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
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0answers
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 ...
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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
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1answer
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 \\ ...