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for questions on algebraic geometry, including algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology.

1
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0answers
48 views

Noether-Lefschetz for Kummer-quartic surfaces with curves of bounded degree

The Noether-Lefschetz theorem states, that a very general quartic surface $S$ in $\mathbb{P}^3$ has $Pic(S)\cong\mathbb{Z}$ generated by $\mathcal{O}_{\mathbb{P}^3}(1)_{|S}$, so all curves on $S$ are ...
2
votes
1answer
67 views

Ideal of the union of two zero loci

Let $X$ be a smooth (complex) projective variety and $\mathcal E$ a globally generated vector bundle on $X$ of rank $< dim(X)$. I would like to know, please, if there is a nice description (exact ...
1
vote
0answers
88 views

Algebraic equivalence of cycles and Chow varieties

Let $p,d\geq 0$ be two integers and let $X\subseteq\mathbb{P}^N$ be acomplex projective variety. Denote the Chow variety of $X$ consisting of $p$-cycles of degree $d$ by $\mathcal{C}_{p,d}(X)$. I'm ...
0
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0answers
25 views

How many points are still contained in a common (hyper-)ellipsoid?

It is known that $${d+2\choose 2}-1$$ points uniquely determine a quadric in $\Bbb R^d$. However, I want my points not on an arbitrary quadric, but on a centered hyperellipsoid in $\Bbb R^d$, or ...
4
votes
0answers
112 views

Twistor projective line

Let $\mathbf{P}^1$ be the projective line over $\mathbf{C}$, and $c : \mathbf{P}^1\to \mathbf{P}^1$ the anti-holomorphic involution sending $z$ to $-\overline{z}$. The quotient $\widetilde{\mathbf{P}}...
1
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0answers
41 views

Direct limit of complexes from cocycle property up to homotopy

Suppose that we have a directed set $(I, \leq)$, and a set of maps \begin{equation} f_{i,j} : C^*(A_i) \to C^*(A_j), \quad i \leq j, \end{equation} of singular cochain complexes of topological spaces $...
7
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0answers
61 views

Can elements in the orthogonal group of a non-split Azumaya algebra with an orthogonal involution have reduced norm -1?

Let $R$ be a connected (commutative) ring with $2\in R^\times$. Let $A$ be an Azumaya algebra over $R$ and let $\sigma:A\to A$ be an orthogonal involution. (This means that there is a faithfully flat ...
3
votes
3answers
170 views

Extending a continuous map over projective space

Let $X = P^{n-1}(\Bbb C)$ ($(n-1)$-dimensional projective space) with $n \geq 3$, and let $K \subset X$ denote a compact subset. I have a bijective, continuous map $\phi:K \to K$ which satisfies the ...
1
vote
0answers
156 views

Degeneration of relative Hodge-de Rham spectral sequence

$$\require{AMScd}$$ $$\newcommand{\CC}{\mathbb{C}} \newcommand{\RR}{\mathbb{R}} \newcommand{\Hdr}{H_{\mathrm{dRh}}} \newcommand{\tensor}{\otimes} \newcommand{\Ohol}{\mathcal{O}}$$ Please excuse that ...
2
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0answers
35 views

Ranks of cycles with base field coefficients as a generalization of ranks of multivectors?

This must be probably only reference request since I am inclined to believe that I am asking about something well known but just cannot pin down appropriate keywords for searching. The starting point:...
3
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0answers
97 views

Confusion in known result about Moduli space of vector bundle of rank 2 degree 0 vector bundles over smooth curve of genus 2

Theorem: Let $X$ be a complete, non-singular algebraic curve of genus $2$. Let $U(2, \Theta)$ be the space of $S$-equivalence classes of semi-stable vector bundles of rank $2$ and degree $\Theta$. The ...
8
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2answers
581 views

The Stacks project

I have a question concerning the admirable Stacks Project. Which comparable projects are there: approach-wise: "an open source textbook on algebraic stacks and the algebraic geometry that ...
3
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0answers
86 views

Reference Request: Fourier Mukai on non Weierstrass Elliptic Fibration

I'm aware of the standard results about Fourier Mukai on Weierstrass elliptic fibration. What I need is the references about Fourier Mukai on elliptic fibration which can have reducible fibers and/...
7
votes
1answer
288 views

What are some open problems in moduli spaces and moduli stacks?

I would like to know what are the open big and interesting problems related to moduli spaces and moduli stacks ? Thanks in advance for your help.
5
votes
2answers
324 views

A paradox on the deformation of singularities

Setup: $\pi: \mathcal X \to C$ is a flat morphism from a germ of a smooth curve $(C, o)$. Suppose the special fiber $\pi^{-1}(o) := \mathcal X_o$ has a certain class of singularities, one wants to ...
4
votes
1answer
205 views

A question on the slice filtration and the slice of the motive of the projective space

In the following $k$ is an algebraically closed field of characteristic $0$. Consider the category $SH(k)$ (the Morel-Voevodsky stable motivic homotopy category). By the work of Voevodsky (see for ...
1
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1answer
141 views

Subschemes of projective varieties

I'm studying the article Introduction to Lawson Homology of Peters and Kosarew. In their exposition of Lawson homology they call projective variety any zero-locus $X$ of homogeneous polynomials in ...
10
votes
1answer
586 views

Books with exercises to learn Langlands program, Galois representations, modular forms

I want to develop a basic background in number theory with a goal towards contributing to some part of the Langlands program (which I know is vast, and has many different aspects; I want to do ...
8
votes
0answers
180 views

On a paper of Formanek about $PGL_4$

In his paper "The center of the ring of 4 × 4 generic matrices" (Journal of Algebra, 1980) Formanek shows that, if $V$ is the representation of $PGL_4$ given by simultaneous conjugation of pairs of ...
1
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0answers
80 views

When does the Jacobian of a smooth curve contains an unique principal polarization

Let $X$ be a smooth, projective curve of genus at least $4$ and $X$ non-hyperelliptic. I am looking for additional conditions on $X$ such that the Jacobian $J(X)$ of $X$ contains an unique principal ...
2
votes
2answers
185 views

Intuition behind Kawamata's definition of a relative movable Cartier divisor

I am trying to develop a good geometric intuition and to understand the motivation behind Kawamata's definition of a relative movable Cartier divisor in Section 2 of reference [1]: [1] Y. Kawamata, ...
2
votes
1answer
120 views

Function is $L^p$-integrable for $p >1$ [Kähler Geometry]

I am reading through a proof in W. Ding and G. Tian's 1992 paper on the generalised Futaki invariant. To provide context, we are looking for obstructions to the existence of Kähler--Einstein metrics ...
7
votes
2answers
354 views

Understand the difference between two stacks

Let us work over $\mathbb{C}$. Let $G$ be a finite group, acting on $\mathbb{A}^1$ via a character, and let $H$ be the kernel of the action. Assume that $\mathbb{A}^1$ is the coarse moduli space of ...
2
votes
1answer
157 views

Naive question on the Jacobian of a curve

Let $X$ be a smooth, projective curve of genus $g \ge 2$. We know that the Jacobian $J(X)$ of the curve is a principally polarized abelian variety. The principal polarization is induced by the ...
1
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0answers
73 views

Fiber product of an elliptic surface

Let $f:S\to P^1$ be a smooth elliptic surface and let $X=S\times_{P^1} S$ be the fiber product. The threefold $X$ is singular in general (typically isolated ODPs). But is $X$ $\mathbb Q$-factorial? Or,...
7
votes
1answer
184 views

Do arithmetic schemes have non-singular alterations?

Let $X$ be an integral normal flat finite type scheme over $\mathbb{Z}$. Does there exist a proper surjective generically finite morphism of schemes $Y\to X$ with $Y$ an integral regular ...
1
vote
1answer
91 views

Lifting Lang-Steinberg to DVR's in Characteristic 0

Let $A$ be a compact DVR in characteristic $0$, uniformizer $\pi$ and residue field $k$. Let $A\subset B$ be a complete DVR with the same uniformizer $\pi$ and algebraicly closed residue field $F$. ...
2
votes
0answers
192 views

Hodge conjecture and K-theory

I am trying to understand the following statement from Gillet book. If someone could provide me reference for that. "The hodge conjecture can be understood as assertions about the image of the natural ...
2
votes
0answers
50 views

Blowing up the base of an elliptically fibered (non Weierstrass) threefold

Suppose $X$ is an elliptically fibered smooth threefold, with a nontrivial Mordel-Weil group. Lets call the sections $\sigma_i$ ,$i=1 \dots n$. None of these "sections" are honestly a section, they ...
6
votes
0answers
81 views

Can the base of an elliptically fibered Calabi-Yau threefold be an Enriques surface?

For this question, a Calabi-Yau manifold or variety of dimension $n$ is defined as a non-singular projective variety with trivial canonical bundle and $h^{i,0} = 0$ unless $i = 0$ or $i = n$. If ...
5
votes
0answers
74 views

Good reduction of abelian varieties over valuation rings via coverings

Let $\mathcal{O}_K$ be a valuation ring with fraction field $K$, and let $A$ be an abelian variety over $K$. Suppose that there is a smooth proper scheme $\mathcal{X}$ over $\mathcal{O}_K$ whose ...
8
votes
2answers
226 views

Berkovich space including both archimedean and non-archimedean worlds

From this Temkin's paper (at the end of section 1.1.3), I know that one may define Berkovich spaces that include both archimedean and non-archimedean worlds. This looks very interesting. Temkin ...
9
votes
2answers
291 views

Do abelian varieties have Neron models over arbitrary valuation rings?

Let $\mathcal{O}_K$ be a valuation ring with fraction field $K$. Let $A$ be an abelian variety over $K$. Does $A$ have a Neron model? If $\mathcal{O}_K$ is a discrete valuation ring, then this is ...
10
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0answers
111 views

Curves on rational surfaces and Lang's conjecture for M_g

There are a group of related conjectures associated to Lang's name - for this question I'll consider only the weakest one, namely that rational curves in a projective variety of general type are not ...
3
votes
0answers
62 views

Is there Thom isomorphism for equivariant K groups in algebraic geometry, not necessarily complex number field?

In Chriss and Ginzburg's fantastic book 'representation theory and complex geometry', they use the following Thom Isomorphism: $\pi:E\rightarrow X$, is a G-equivariant affine linear bundle, then $\pi^...
0
votes
0answers
88 views

About the multiplicative group of p-adic complex

I was studying the multiplicative group of the $\mathbb{C}_p$. I'm interesed in the ring $\mathcal{O}_p$ of elements in $x\in\mathbb{C}_p$ such that $|x|_p\geq 1$. I have three questions. The first ...
0
votes
0answers
59 views

Computing algebraic entropy

Could you recommend any reference for computing algebraic entropy? Here algebraic entropy is defiened as $\lim_{n \to \infty}\log (deg (f^n))^{1/n}$ for a rational map $f $. I saw that there are ...
3
votes
0answers
587 views

Work of Caucher Birkar

I am asking this since the work of this Fields medallist was not covered in the related question on work of 2018 ICM plenary speakers below. Work of plenary speakers at ICM 2018 Terry Tao has some ...
5
votes
0answers
86 views
+50

Algebraic independence of limit cycles of Lienard equation

It is well known that the Van der Pol equation $$\begin{cases} \dot x=y-(x^3-x)\\\dot y=-x \end{cases}$$ has no an algebraic limit cycle. According to this fact, we search for a related ...
3
votes
0answers
109 views

Cohomology of the Hilbert square of a degree 14 K3 surface [Beauville-Donagi]

I have a question about the article by Beauville-Donagi called La variété des droites d'une hypersurface cubique de dimension 4 (C. R. Acad. Sc. Paris, t. 301, Série I, n° 14, 1985). Their ...
5
votes
1answer
241 views

Degeneration of smooth curves and Picard-Lefschetz formula

Let $\pi:\mathcal{C} \to \Delta$ be a family of projective curves of genus $g \ge 2$ over the unit disc $\Delta$, smooth over the punctured disc $\Delta\backslash \{0\}$ and central fiber $\pi^{-1}(0)$...
7
votes
0answers
148 views

Explicit descriptions of a flop

I want to know how to describe explicitly the flop of the following flop contraction. Because the construction is so natural and simple, I was wondering such descriptions should already exist in the ...
5
votes
3answers
279 views

Smoothen a nodal curve

Let $k$ be a field, let $X/k$ be a nodal curve (which means $X_{\overline{k}}$ is a connected reduced proper curve with at worst nodal singularities.) Does there always exists a proper flat family $\...
26
votes
2answers
2k views

How to visualize Dirichlet’s unit theorem?

As the question title asks for, how do others "visualize" Dirichlet’s unit theorem? I just think of it as a result in algebraic number theory and not one in algebraic geometry. Bonus points for ...
7
votes
3answers
245 views

Infinite Galois descent for finitely generated commutative algebras over a field

Let $k_0$ be a field of characteristic 0, and let $k$ is a fixed algebraic closure of $k_0$. Write $G={\rm Gal}(k/k_0)$. Let $A_0$ be a finitely generated commutative $k_0$-algebra with a unit. Then ...
0
votes
2answers
149 views

How to find the normal form of an elliptic curve?

Let $C$ be the following curve in $\mathbb{C}^2$. \begin{align} & 11664\, {c_1}^3\, {c_2}^2 + 536544\, {c_1}^3\, c_2 + 6170256\, {c_1}^3 + 67068\, {c_1}^2\, {c_2}^2 + 1542564\, {c_1}^2\, c_2 \\ &...
0
votes
1answer
127 views

How to classify a plane complex curve?

Let $p_1, p_2, t_1, t_2, a \in \mathbb{C}$ be constants. Consider the following plane complex curve in $\mathbb{C}^2$ ($c_1, c_2$ are indetermniates) \begin{align} & {p_1}^2 {p_2}^2 c_1 {t_1}^2 ...
1
vote
0answers
45 views

Branch multipicity of curves

Say $C_1$ and $C_2$ are curve classes on a smooth projective complex surface $X$. (In my case, $X$ is toric.) Let $P$ be a point on $X$. Let $D_i$ be a divisor in the class $C_i$, and let $m_i$ ...
0
votes
0answers
23 views

critical points relevant to the lowest order non-perturbative correction

I am interested in the Hyperasymptotics of multidimensional integrals of the form $$\mathcal{I}(\lambda) = \int_{\mathbb{R}^n} dz_1 \wedge dz_2 \wedge \dotsi \wedge dz_n \, g(z_1,\dotsi,z_n) \, e^{\...
2
votes
0answers
90 views

Pushforward of structure sheaf on quotient surface singularity

Let $f:\mathbb{C}^2 \to \mathbb{C}^2/G$ be a quotient surface singularity. What properties should $G$ have such that the pushforward $f_*\mathcal{O}_{\mathbb{C}^2}$, of the structure sheaf $\mathcal{O}...