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

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On orders of stabilisers of group actions and stacks

Let $X$ be a finite type irreducible separated DM-stack over $\mathbb C$. Let $x$ be an object of $X(\mathbb C)$ with stabilizer $G_x$. Let $y$ be an object of $X(\mathbb C)$ with stabilizer $G_y$. ...
3
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1answer
177 views

Surjectivity of the Kodaira-Spencer map

Let $X$ be a complex projective manifold. Let $B$ be the closed subscheme of $H^1(X,T_X)$ defined by $\mathfrak{m}^2$, where $\mathfrak{m}$ is the ideal defining the origin. In other words, $B$ is a ...
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64 views

Rank growth in ray class fields of primes that are inert in an imaginary quadratic extension

If $K=\mathbb{Q}[\sqrt{-d}]$, $E$ is an elliptic curve and $\operatorname{rank}(E(K))=\operatorname{rank}(E(\mathbb{Q}))$, are there infinitely many primes $l$ of $\mathbb{Q}$ so that: (1.) $\...
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1answer
91 views

Does the integral closure of a normal local ring in a finite extension of its fraction field have finite projective dimension?

Let $A$ be a normal local domain with field of fractions $K$. Let $L$ be a finite separable extension of $K$ (if relevant, I'm happy to assume all possible ramification is tame), and let $B$ be the ...
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91 views

Lifting sections to completion of closed subschemes

Let $R$ be a reduced finite type $\bar{k}$-algebra, a projective morphism $\pi \colon V \rightarrow \mathrm{Spec}(R)$ and ideals $I, J \subseteq R$. Assume there is a split $s_{IJ} \colon \mathrm{Spec}...
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557 views

Are there nonlinear projective spaces?

This is actually a series of questions posed by Guram Berishvili about the structure he calls marao. Everything I am going to write here I took (and messed up) from his home page which is all in ...
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128 views

Smooth quotients of algebraic spaces that are varieties away from codimension $\ge 2$ subset

This is a question about when a smooth complex algebraic space that is very close to being an algebraic variety is actually an algebraic variety. General question: Let $X$ be a smooth separated ...
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118 views

Moduli of hyperelliptic curves: odd vs even genus

I'm stumped by Exercise 2.3 in Harris-Morrison, which says: "Show that there does not exist a universal family of curves of genus 2 over any open subset $U \subset M_2$. In general, if $H_g \subset ...
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82 views

Riemann surface defined by a Beltrami differential

Let $R$ be a Riemann surface, $\omega$ and $\tau$ respectively a holomorphic and an antiholomorphic 1-form on $R$. Locally $\omega=fdz$ and $\tau=gd\overline{z}$ with $\partial_{\overline{z}}f=0$ and $...
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65 views

The pull-back of two cycles of different dimension which are linearly equivalent

Let $p:X\rightarrow {\rm Grass}(k,n)$ be a generically finite-to-one surjective morphism from an algebraic variety to the complex grassmannian ${\rm Grass}(k,n)$. If $D$ and $E$ are two reduced and ...
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132 views

Arakelov divisors and the meaning of real coefficients

I'm learning Arakelov theory on arithmetic surfaces and I have the following general question. Let $K$ be a number field and consider its ring of integers $O_K$. Moreover let $S:=\operatorname{Spec} ...
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192 views

Is it clear that $y^3=f(x)$ has bad reduction at $3$?

Bad reduction is defined as 'nonexistence' of a model where the curve has good reduction. So let's take the curve $C$ which is affinely given by $$y^3 = f(x)$$ (absolutely irred, $f$ no multiple roots)...
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58 views

etale projection of the tropicalization

Let $(K,v)$ be a complete valued field with value group $\mathbb R$, and let $X$ be a closed subvariety of dimension $d$ of the algebraic torus $(K^\times)^n$. Let $\sigma$ be a polyhedron in the ...
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76 views

Counting points in a certain 4-dimensional region

Let $(a,b,c)$ be a fixed tuple of co-prime integers, with $a \ne 0$ and at least one of $b,c$ non-zero. Define $$L = -\frac{a p_1 q_1 - b p_2 q_1 - b p_1 q_2 + 4 c p_2q_2}{a},$$ $$Q = \frac{(b^2 - ...
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205 views

Why are formal schemes assumed to be (locally) noetherian?

All sources that I know that study formal schemes seem to assume that they are locally noetherian. For instance, in Hartshorne "Algebraic Geometry", the author states: "For technical reasons we will ...
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335 views

Higgs fields whose determinant have only simple zeros

Is the following property true for every stable holomorphic bundle of rank 2 with trivial determinant on a compact Riemann surface: The space of trace-free Higgs fields, whose determinant have only ...
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192 views

A mysterious quasi-isomorphism in Kashiwara-Schapira's proof of HKR

On p. 127 of Kashiwara-Schapira's paper "Deformation Quantization Modules", there is the following situation: $X$ is a smooth complex (quasi?)projective variety and $\delta\colon X\to X\times X$ is ...
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1k views

Where am I suppose to actually learn how to compute hypercohomology?

I'm reading about algebraic de Rham cohomology over characteristic zero which is constructed using hypercohomology. Already, constructing injective resolutions is difficult, and coupling this with ...
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1answer
166 views

Intersection of two curves is not Cohen Macaulay

Let be $R=\mathbb{C} \lbrace x,y,z \rbrace$ the formal series ring and let $f_{1},f_{2},f_{3} \in R$ be nonzero elements of $R$. (a) Consider the varieties $M:=V(f_{1},f_{2})$ and $N:=V(f_{2},f_{3})$ ...
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205 views

Some elementary questions about deformation quantization

I am interested in deformations of affine Poisson algebras, and so this is the setting in which I shall write out the elementary definitions involved. All algebras and vector spaces shall be over $\...
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226 views

The derived version of the Grothendieck spectral sequence

Consider the (very well known) Grothendieck spectral sequence for composition of functors $\mathcal F: \mathcal A \to \mathcal B$ and $\mathcal G: \mathcal B \to \mathcal C$ between abelian categories ...
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113 views

Singular Locus of a Schubert variety

I am trying to compute the singular locus of the schubert variety $X_w$ in $G_{2,7}$ where $w=(4,7) \in I_{2,7}$. Following the notation in the book "The Grassmannian Variety: Geometric and ...
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119 views

Pseudoeffective line bundle

Let $L$ be a pseudoeffective line bundle on a complex manifold $X$ then is there a singular hermitian metric of $L$ which its curvature is not semi-positive?
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176 views

Equivalence of algebraic and topological monodromy representations?

Does anyone know of a reference for the following fact? Let $M_g$ denote the moduli stack of genus g curves, let $A_g$ denote the moduli stack of abelian varieties, and let $U_g \rightarrow A_g$ ...
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1answer
78 views

How to compute graph ideal or cut ideal of a graph?

Graph ideals are a special case of Stanley-Reisner ideal, explained in Combinatorial Commutative Algebra book by Sturmfels, and graph ideals here. Graph ideals are generated by the minimal paths while ...
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1answer
109 views

Criterion for a polynomial ideal to be radical

Let $K$ be an algebraically closed field, and let $I$ be an ideal of $K[x_1,\dots,x_n]$ ($n \geq 2$). We suppose that for all $i$, and for all $a_1,\dots,a_{i-1},a_{i+1},\dots,a_n \in K^{n-1}$, the ...
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107 views

Quotient of Dedekind domains

Is there any characterization for a commutative ring to be a quotient of a Dedekind domain?
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1answer
67 views

Cut ideal of two graphs?

Consider a connected graph $G$ and a connected graph $H$. Their graph ideals are their path ideals. The Alexander duality of the graph ideals give the cut ideals. $G$ and $H$ are not connected to each ...
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100 views

singular points of $\alpha_{p}$-torsor and $\mu_{p}$-torsor of curves

Let $X$ be a smooth projective curve over an algebraically closed field of characteristic $p>0$. Suppose that the genus of $X$ is >2. Let $Y$ be a non-trivial $\alpha_{p}$-torsor or $\mu_{p}$-...
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184 views

Do finite flat sheaves define families of $0$-cycles?

Let $X$ be a smooth projective $\mathbb C$-variety and let $X^{(n)}$ denote the symmetric product $X^n/S_n$, parametrizing effective $0$-cycles of degree $n$ on $X$. Question. Let $S$ be a ...
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233 views

How often does a pair of linear maps generate a Zariski-dense subgroup of $GL(d,\mathbb{R})$?

I am an analyst working on a number of problems which in some way relate to random matrix products. In this context I frequently find that the analytic properties I am interested in depend in some way ...
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1answer
141 views

Bimodules of fractions

This question is about extending modules of fractions to bimodules of fractions. I would not be surprised if the result is known, I have tried looking in Goodearl and Warfield, but may have missed the ...
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1answer
188 views

A property of minimal prime ideals in commutative reduced ring

Let $\{\mathfrak{p}_i\}_{i\in I}$ ($I$ is an infinite set) be a family of minimal prime ideals in a commutative reduced ring $R$ with identity, and let $a, b \in R$. If the ideal $\langle a, b\rangle$...
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158 views

on universal homeomorphisms between schemes

We are taught since when we are young that schemes are cool because they take into account "nilpotents". This means also that we can distinguish between schemes which have the same underlying ...
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30 views

Tropical self intersection number of boundary divisor on toroidal embedding

Let $X_0 \subset X$ a toroidal embedding without self intersections and denote by $\overline{\Sigma}$ its corresponding (weakly embedded) extended conical simplicial complex. Let $D$ be a divisor on $...
4
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1answer
225 views

How to construct an abelian variety with CM by a given CM field?

Let $F$ be a totally real number field, and let $K$ be a quadratic extension of $F$ which cannot be embedded into $\mathbb{R}$. Then $K$ is a so called CM field. For instance, take $F = \mathbb{Q}(\...
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23 views

Property of solid spanned by optimal functions

I can only solve this in a two-dimensional world. I want to characterize a solid that is spanned by cutoff functions $g$ that are optimal in the following sense. Let us suppose we have $n$ random ...
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213 views

On algebraic morphisms

Let given schemes $Y\subset X$ and $Z$, where $Y$ is closed subscheme of $X$. Assume that for some morphism $f:Y\to Z$, $Z = [f(Y)]$ (where [] means closure in $Z$). Is it true that there exists ...
10
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1answer
239 views

Why is there a factor $p$ in the definition of $T_p$ via Hecke correspondences on modular curves?

Fix $N\ge4$. Let $Y_1(N)$ and $X_1(N)$ be the usual modular curves. I want to view them as schemes over $\mathbb Z$ representing the moduli functors of (usual or generalized) elliptic curves with (...
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1answer
143 views

Does a moving family of lines through a fixed point produce a singularity?

This is just a feeling that I had and I am curious if it is totally wrong or true to some extent. Let $X\subseteq \mathbb{P}^r$ be an integral hypersurface of degree $r-1$, which is not a cone. In ...
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1answer
132 views

Sequences of divisors satisfying Serre vanishing?

Serre's vanishing theorem (SV) states that, on a projective variety $X$ with a choice of ample line bundle $\mathcal{O}_X(1)$, for any coherent sheaf $F$, we have $$H^i(X,F(m))=0,\quad m>>0$$ ...
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86 views

Bounding the number of “generalized $\mathbb{F}_q$-rational points” of a variety in terms of dimension and degree

In what follows, for a prime power $q=p^m$, $\phi_q$ denotes the Frobenius endomorphism $x\mapsto x^q$ of a finite-dimensional affine space over the algebraic closure $\overline{\mathbb{F}_p}$ (the ...
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101 views

A sufficient condition for a morphism to be a closed immersion?

Let $R$ be an integral $\bar{\mathbb{F}}_p$-algebra of finite type, let $V$ be an $R$-algebra. Consider a morphism $f \colon \mathrm{Spec}(V) \rightarrow \mathbb{A}^n_R$ that has the following ...
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Is a Kummer surface unirational over a sufficiently large finite field of characteristic 2?

Let $A$ be a supersingular abelian surface over a sufficiently large finite field $\mathbb{F}_q$ of characteristic $2$ and let $K_A = A/(-1)$ be the Kummer surface. Shioda ("Kummer surfaces in ...
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117 views

Are two conic bundles birational, if their bases are birational via a map preserving the associated quaternion algebras?

Assume we have two standard conic bundles $\pi:C \rightarrow X$ and $\pi': C'\rightarrow X'$. That is $\pi$ and $\pi'$ are flat morphims of smooth varieties over $\mathbb{C}$ and both maps are ...
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96 views

GIT: For $x$ fixed, is $\{L:x \in X^s(L)\}$ open in $\text{Pic}^G(L)$?

Let $G$ be a complex reductive algebraic group acting on a complex variety $X$ (not necessarily projective) with $\text{Pic}^G(X)$ finite dimensional (for simplicity). For a fixed $x \in X$ define $$P^...
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69 views

Function field of a Drinfeld module, product formula, or Chow group

I am learning about Drinfeld modules, and I have a few questions. There is an analogue that Drinfeld modules are like elliptic curves, which are projective, or are compact Riemann surfaces over $\...
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87 views

is the “reduction” of an effective cartier divisor on a relative curve still a cartier divisor?

Let $C\rightarrow S$ be a smooth proper morphism of relative dimension 1, where $S$ is a Noetherian normal scheme. Let $D\hookrightarrow C$ be a relative effective Cartier divisor finite over $S$. Let ...
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1answer
133 views

Riemann-Roch for reducible surfaces

Let $C$ be a projective connected (reducible) curve over an algeraically closed field with nodes as singularities and $X=\mathbb P(\mathcal E)$ a projective bundle over $C$ (we know a ...
4
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1answer
382 views

Three dimensional representations of Alternating group

The alternating group $A_5$ has $2$ irreducible representation of degree $3$. The characters for these representations have irrational values. I guess the ring of invariants of these representations ...