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

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264 views

Picard functor of an algebraic group

Let $K$ be a field, $G$ a $K$-group scheme of finite type, and $X$ a $G$-torsor. Is the Picard functor $\mathrm{Pic}_{X/K}$ representable? I recall that $\mathrm{Pic}_{X/K}$ is the fppf sheafification ...
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1answer
119 views

Extending descent data from the special fiber of an extension of DVR's

My question is about the proof of Lemma D.3 on p. 147 of the book "Neron models" by Bosch, Lutkebohmert, and Raynaud. Namely, towards the end of that proof there is the sentence "That $\varphi$ ...
2
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2answers
205 views

Closure relations between Bruhat cells on the flag variety

Given a Lie group $G$ over $\mathbb{C}$ and a Borel subgroup $B$. There is this famous Bruhat decomposition of the flag variety $G/B$. How do we prove the closure relations between the cells, which ...
7
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1answer
289 views

Computing naive algebraic singular homology

I'm not sure if this counts as research level, since it might just be an expression of my ignorance. But anyway. Let $\Delta^n_A = Spec(A[X_0, \dots, X_n]/\sum X_i - 1)$ denote the standard algebraic ...
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3answers
271 views

Why there are two point at infinity on certain elliptic curve [closed]

In article Adams, W. W., & Razar, M. J. (1980). Multiples of points on elliptic curves and continued fractions. Proc. London Math. Soc, 41, 481-498. is said on ...
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4answers
488 views

Good lecture notes/books on Jacobian of hyperelliptic curve

I want to understand what the Jacobian variety is from an algebraic (or arithmetic?) perspective. I want to know: What is the definition of the Jacobian? Widely know facts about it. Why the ...
7
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1answer
205 views

Rationality of moduli spaces of rational curves

Let $\overline{M}_{0,n}$ be the moduli space of Deligne-Mumford stable pointed rational curves, and let us consider the quotient $\widetilde{M}_{0,n} = \overline{M}_{0,n}/S_n$. Clearly, there is a ...
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2answers
133 views

Matroids relaxations of a given matroid

Let $\mathcal{M}$ be a rank-$d$ matroid on $[n]$. Say a matroid $\mathcal{N}$ is a relaxation of $\mathcal{M}$ if $\mathrm{rank}(\mathcal{N})=d$, $\mathrm{groundset}(\mathcal{N})=[n]$, and every ...
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2answers
160 views

Connected unipotent algebraic groups

Let $G$ be a connected unipotent algebraic (affine) group defined over a perfect field $k$. Then $G$ is $k$-isomorphic as an algebraic $k$-variety to the affine space $\mathbb A^n_k$. Is there any ...
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85 views

Dimensions of two spaces

Let $R=\Bbb K[x_1,\dots,x_n]/I$ and $Z=\mathsf Z(I)$ where $\Bbb K$ is any field with $char(\Bbb K)\neq 2$. Is there a way to describe space of $f_1,f_2\in R$ that satisfies $$f_1+f_2\neq 0,\mbox{ ...
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1answer
153 views

About Prop 1.1 of “…Petri's Analysis…” by Stöhr & Viana

I'm reading this paper and the authors Karl-Otto Stöhr and Paulo Viana define $\Omega^n$ to be the set of the sum of the monomios $\omega_1\cdots\omega_n$, where $\omega_i\in \Omega$ and $\Omega^n(D)$ ...
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1answer
102 views

Regular rings and formally smooth algebras

Let $A\rightarrow B$ be a commutative $A$-algebra. If $A$ is a field and $B$ Noetherian and formally smooth over $A$, then it is known that $B$ must be a regular ring. Is there a partial converse of ...
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1answer
259 views

Concrete Examples of Shimura Surfaces

First a disclaimer: I am at best a part-time arithmetic geometer, so please accept my apologies when I am too naive or get something wrong. From time to time I have tried to learn something about ...
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75 views

inducing group action on a blowup

Let $X$, $Y$ be a smooth projective varieties, say over the complex numbers, both acted upon by a connected linear group $G$. Let $f\colon X\to Y$ be an equivariant rational map. Let $Z$ be a smooth ...
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1answer
151 views

How many circular cylinders through 5 general points?

How many right circular cylinders can pass through 5 general points in ℝ3 ? Edit: 0,2,4, or 6.
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2answers
268 views

Generalization of Pascal's Theorem to Higher Dimensions

Pascal's celebrated theorem in classical geometry gives a necessary and sufficient condition for the existence of a conic through six given points in the plane. Does there exists a similar statement ...
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0answers
79 views

Is there a Kummer theory for cyclic covers in higher dimension?

Let $d \geq 2$ be an integer and $K$ a field of characteristic zero containing the roots of unity of order $d$. Let $X$ be a smooth curve over $K$, not necessarily projective. Let $\overline{x}$ be a ...
3
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1answer
179 views

Etale Realization and Gysin Sequence

Ivorra defined a tensor triangulated functor from Voevodsky's triangulated category of motives to the derived category of complexes of etale sheaves of $\mathbb{Z}/n$ modules with bounded cohomology ...
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66 views

Homology class of variety defined by an ideal

if a subvariety of codimension n is given by an ideal of polynomials with n generators, then the homology class of the variety is given by the intersection product of the classes of the individual ...
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2answers
640 views

Is there a scheme parametrizing the closed subgroups of an algebraic group?

In the following, let $G=\operatorname{GL}_n(\mathbb{C})$ or $G=\operatorname{\mathbb PGL}_n(\mathbb{C})$, depending on whichever has a better chance of yielding an affirmative answer. One could more ...
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0answers
102 views

How big can a family of pairwise intesecting affine spaces be?

I apologize if this question might seem to be a bit too elementary. Let $\mathbb{P}^n$ be the projective space over $k$ - an algebraically closed field of characteristic 0. Let $1\leq l\leq n-1$, and ...
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104 views

residue and regulator

Let $C$ be a curve defined over $\mathbb{Q}$. The regulator is a map $$ reg: K_2(C)_{\mathbb{Q}} \longrightarrow H^1(C(\mathbb{C}), \mathbb{R}). $$ Here $K_2(C)_{\mathbb{Q}}$ is the K-group tensor ...
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3answers
192 views

Characterizing orthants with polynomials

Let $x=(x_1,\ldots,x_n)\in\mathbb{R}^n$. Can one find a polynomial $p$ (of arbitrary degree) in the coordinates of $x$ such that $p(x)\geq 0$ if and only if $x$ is an element of the positive orthant ...
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79 views

Closed Invariant Forms on Complex Projective $k$-Space

Considering complex projective $k$-space as the homogeneous space $SU_k/U_{k-1}$, is it true that every $SU_k$-invariant form is closed?
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2answers
470 views

Adjoining torsion points from abelian varieties

Let $L/\mathbb{Q}$ be the field generated over $\mathbb{Q}$ by all of the (projective) coordinates of all of the torsion points of all abelian varieties defined over $\mathbb{Q}$. Is $L$ algebraically ...
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0answers
68 views

Equidimensionality of stalks of $\operatorname{Proj} S$ when $S$ is equidimensional.

I would like to know a reference of the following statement (or counter example). Let $S$ be a (commutative) Noetherian standard graded ring over a local ring, i.e., $S = S_0[S_1]$, where $S_0$ is ...
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1answer
150 views

Finite generation of certain $\mathcal{O}_X$-algebra

It is proved in this paper by Kawamata (Theorem 6.1) that for a 3-dimensional normal algebraic variety $X$ which has at most canonical singularities, and a Weil divisor $D$ on it, the ...
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214 views

Am I missing something about this notion of Mirror Symmetry for abelian varieties?

This is a continuation of my recent question: Mirror symmetry for polarized abelian surfaces and Shioda-Inose K3s. In the comments of the question, I was directed to the paper ...
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0answers
88 views

“Non-symmetric” Polarized Hodge Structure?

In the definition of a polarized Hodge structure (see here for a definition) is the assumption that the generalized Hodge--Riemann pairing is symmetric up to a possible sign. Does anybody work with ...
3
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1answer
229 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 ...
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0answers
104 views

Solution to system of polynomial equations

Suppose $A$ is a set of polynomials:$$P_1(x,y_1,\dots,y_n)=0,$$ $$P_2(x,y_1,\dots,y_n)=0,$$ $$\vdots$$ $$P_k(x,y_1,\dots,y_n)=0$$ is a system of equations with coefficients over $\mathbb{Z}$, and ...
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0answers
87 views

A question on the secondary fan

I am studying the secondary fan decomposition of the effective cone of a projective variety $X$. Let as assume that $X$ is a Mori Dream Space. As far as I understand passing from a cone of maximal ...
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1answer
130 views

Cartier divisor on a double cover

Suppose $X$ is a projective variety, and $D$ is a Cartier divisor on $X$. Is it possible (or this is always true) that there is a (ramified) double cover $\pi: \tilde{X} \to X$ such that ...
2
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1answer
333 views

Symplectic quotient of projective variety is projective?

Let $G$ be a compact connected Lie group and $\mathfrak g^*$ be dual of Lie algebra $\mathfrak g$. Let $M$ be a compact projective variety and $G$ act on $M$ freely and $M$ is $G$ equivariant, and ...
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0answers
216 views

Affine communication lemma and finite limits in the category of rings

Let $X$ be a scheme and $\mathrm{Spec}(B) = V \subseteq X$ be an open affine subset. When using the affine communication lemma (c.f. Theorem 6.3.2, Vakil's notes, Foundations of Algebraic Geometry), ...
4
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1answer
165 views

Intersections of hypersurfaces of degree $d$ in $\mathbb CP^n$

Is it true that every projective sub-variety of degree $d$ in $\mathbb CP^n$ is an intersection of some number of hypersurfaces of degree $d$? Is there some simple proof of this fact? (I believe this ...
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0answers
194 views

On the Hitchin fibration

I will refer to Simpson's "Higgs bundles and local systems". Proposition 1.4: When $X$ is a smooth projective variety, one can build up the moduli space $\mathcal{M}(X,r)$ of rank $r$ Higgs ...
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102 views

Is this $S$-birational map an open immersion on its domain of definition?

My question is about a claim on the bottom of p. 121 of the book "Neron models" by Bosch, Lutkebohmert, and Raynaud, so I will freely use the general terminology recalled in this book, but will ...
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0answers
89 views

Questions about the Moduli Space of Vector Bundles of rank 2 with trivial determinant [migrated]

Let $M_0$ be the moduli space of rank 2 semi-stable vector bundles over X with trivial determinant which is a singular projective variety of dimension $3g-3$. $M_0$ is constructed as the $GIT$ ...
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1answer
129 views

ideal of maximal minors is cohen-macaulay?

Let $k$ be an algebraically closed field. Let $A$ be an $m \times n$ matrix with linear forms $a_{ij} \in k[x_1, \ldots, x_p]_1$ as entries. Let $I$ be the ideal generated by the maximal minors of ...
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0answers
108 views

Proper monomorphisms in complex analytic spaces

In the algebraic geometry of schemes, we know that monomorphisms which are universally closed (= every pullback is a closed map) and of finite type are closed immersions. See Gortz, Wedhorn "Algebraic ...
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0answers
101 views

How to see that this pairing of line bundles is multiplicative?

Given a projective flat morphism $p: X \rightarrow Y$ of integral noetherian schemes of relative dimension one. For a coherent sheaf $F$ on $Y$ we can define a line bundle $det(F)$ on $Y$ and for a ...
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1answer
1k views

Algebraic dependency over $\mathbb{F}_{2}$

Let $f_{1},f_{2},\ldots,f_{n}$ be $n$ polynomials in $\mathbb{F}_{2}[x_{1},x_{2},\ldots,x_{n}]$ such that $\forall a=(a_1,a_2,\ldots,a_n)\in\mathbb{F}_{2}^{n}$ we have $\forall ...
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1answer
385 views

A funny factorization of the Jacobian coming from the lines on the Fermat cubic

Here is something which came up in my algebraic geometry class, and I'm wondering if it has a deeper explanation. Let $F(w,x,y,z) = w^3+x^3+y^3+z^3$ and let $X$ be the cubic surface in $\mathbb{P}^3$ ...
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4answers
2k views

Fermat's last theorem over larger fields

Fermat's last theorem implies that the number of solutions of $x^5 + y^5 = 1$ over $\mathbb{Q}$ is finite. Is the number of solutions of $x^5 + y^5 = 1$ over $\mathbb{Q}^{\text{ab}}$ finite? Here ...
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142 views

Rational curves and Serre's construction

Why rational curves, used in Serre's construction of vector bundles, usually corresponds to unstable bundle? I saw this affirmation in Richard Thomas's paper on an obstructed bundle on a CY threefold. ...
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39 views

singular point of a complete intersection surface [migrated]

Let $S:= H_1\bigcap H_2\bigcap \cdots \bigcap H_N \subset\mathbb{P} _{\mathbb{C}}^{N+2}$ be a complete intersection surface, where each $H_i$ is a hypersurface defined by a homogeneous equation $f_i$. ...
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0answers
185 views

Etale Slice Theorem

I found the Luna's Slice Theorem very Technical. It will be helpful if someone illustrates the geometry involved in the theorem. Also why this theorem so useful? This is Luna's Slice theorem from a ...
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4answers
3k views

Massive cancellations

Let $A=\{a_1,\ldots,a_k\}$ be a fixed, finite set of reals. Let $S_A(n)$ be the set of all reals that are expressible as the sum of at most $2^n$ terms, where each term is a product of at most $n$ ...
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394 views

Are there known cases of the Mumford–Tate conjecture that do not use Abelian varieties?

(For a formulation of the Mumford–Tate conjecture, see below.) The question As far as I know, all non-trivial known cases of the Mumford–Tate conjecture more or less depend on the Mumford–Tate ...