Questions tagged [ag.algebraic-geometry]

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

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0answers
77 views

Why the trilinear GL_2 model is spherical?

Consider the homogeneous space $X:=GL_2\times GL_2\times GL_2/ H$ where $H=GL_2$ is diagonally embedded into $GL_2\times GL_2\times GL_2$. My question is why $X$ is spherical (i.e., there is a Borel ...
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1answer
232 views

Twisted forms of $\mathrm{SL}(2,q)$

$\DeclareMathOperator\SL{SL}$Let $q = p^r$ be a prime power. Let $H$ denote the subgroup of $\SL(2,\overline{\mathbb{F}}_q)$ consisting of matrices of the form $\begin{pmatrix}a & b\\ b^q & a^...
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128 views

Is there an interesting generalization of Vaserstein's argument for Quillen-Suslin?

One way to see that the algebraic vector bundles on an affine space over a field are trivial is to note that algebraic K-theory is $\mathbb{A}^1$-invariant under a smoothness assumption. Another way ...
13
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2answers
392 views

Does every non-isotrivial 1-parameter family of elliptic curves have a positive rank specialization?

Let $\mathcal{E}/\mathbb{Q}(t)$ be given by $$y^2=x^3+A(T)x+B(T)$$ for some $A(T),B(T)\in\mathbb{Q}[T]$ and assume $\mathcal{E}$ is non-isotrivial (the $j$-invariant $\frac{6912 A(T)^3}{4A(T)^3 + 27B(...
5
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1answer
252 views

Extending rational to integral points

Let $p: \mathcal{X} \rightarrow \text{Spec } \mathcal{O}_K$ be a normal proper Artin stack with finite diagonal. A $K$-rational point is by definition a section $x: \text{Spec}(K) \rightarrow \mathcal{...
4
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0answers
134 views

On a family of maps relating vectors to spinors, which includes a smooth map from $S^3$ to $Q^4$

Let $Cl_n$ denotes the Clifford algebra of $\mathbb{R}^n$ with its euclidean inner product $g$, where $n \geq 3$. We use the following convention: $$ v.w + w.v = 2g(v,w)1.$$ In particular, note that ...
4
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1answer
136 views

Regarding a 'global' version of Chase-Harrison-Rosenberg exact sequence for rings

If $R$ is a commutative ring with identity with a 'nice' action of a finite group $G$, the subring $R^G\subset R$ gives a Galois extension of rings. In this case, S.U. Chase, D.K. Harrison, A. ...
3
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1answer
135 views

Existence of terminal $3$-fold flips

Does there exist a terminal $3$-fold $X$ with a curve $C\subset X$ such that $K_X\cdot C < 0$ admitting a Mori flip $X\dashrightarrow Y$, flipping $C$ to a curve $C'\subset Y$, where the singular ...
1
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1answer
205 views

Interesting examples of direct image bundles

Let $\pi : W \rightarrow Y$ be a holomorphic fibration of complex manifolds. Let $L\rightarrow W$ be a holomorphic line bundle on its total space and denote by $$E^k_q := R^q \pi_*L^k$$ the direct ...
6
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372 views

Proof of Lemma 6.5 in Scholze's Perfectoid Spaces

In the proof of Lemma 6.5(approximation lemma) in Scholze's Perfectoid Spaces, I have the following three questions about $h = f - g^\sharp_c$ and $g^\sharp_{c'}$. (Maybe it's something you'll find ...
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187 views

Non-examples of mixed Tate motives

I was trying to find examples of schemes (preferably smooth) over $\mathbb{C}$ which have motives that aren't mixed Tate. I wasn't able to come up with anything or find an argument that's been written ...
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0answers
155 views

Hermitian metrics/Swan-Serre

We should denote by $M$ a compact topological space and let it's continuous functionals to the complex numbers $\mathbf{C}$ be $C(M)$. By the Swan-Serre theorem there is bijective equivalence between ...
4
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0answers
124 views

Relation between rational Tate module and universal cover of a p-divisible group

We can associate two $\mathbb Q_p$ vector spaces to a $p$-divisible group, and I'm a little confused about the relation between these two groups. First of all, I think part of my problem is that when ...
1
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0answers
99 views

Dimension of the Grassmannian of lines of an hyperplane section

Let $X$ be an isotropic Grassmannian, $Pic(X)=\mathbb Z$ (for example $X$ is a projective space or a quadric hypersurface). Consider a global section $s \in \Gamma(X,L)$, where $L$ is the generator of ...
5
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1answer
275 views

diagonal cubic hypersurfaces

At the end of https://encyclopediaofmath.org/index.php?title=Cubic_hypersurface#References it is stated that the diagonal cubic hypersurface $$ \sum_{i=0}^{2m+1} a_i x_i^3 = 0, m\ge 2 $$ (and ...
2
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0answers
118 views

Existence of numerically trivial classes in the algebraic $K$-theory of a threefold with nontrivial Chern characters

This is a follow-up question to my previous post. Let $X$ be a complex smooth projective variety of dimension $d$. Let $K(X)$ denote the Grothendieck group of coherent sheaves on $X$. There is an ...
5
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1answer
251 views

First Chern class of torsion sheaves

Let $X$ be a smooth projective variety, $\mathscr T$ a torsion sheaf with irreducible support of codimension $1$, say $Z$. Then the first Chern class $c_1(\mathscr T)$ is of form $r[Z]$. Is there ...
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0answers
39 views

Real (non-complex) Du Val singularities for quartics of high global Milnor number

I posted this in MathStackexchange and was advised to come here. As I am only looking for examples, I didn't feel MathOverflow was necessary. I am looking for examples of specific quartic projective ...
5
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0answers
175 views

Non identifiable polynomials

A homogeneous polynomial $F\in\mathbb{C}[x_0,\dots,x_n]_d$ of degree $d$ is $r$-identifiable if it can be written as $$F = l_1^d + \dots +l_r^d$$ in a unique way, where the $l_i$ are linear forms. ...
4
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1answer
226 views

Six functor formalism for quasi-coherent $D$-modules

Let $X$ be a smooth scheme over a field $k$ and let $\mathsf{D}_{\text{qc}}(\mathcal{D}_X)$ be the full subcategory of $\mathsf{D}(\mathcal{D}_X\mathsf{-Mod})$ composed of the complexes of $\mathcal{D}...
3
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0answers
60 views

Poincare polynomials for Borel Moore homology and fibrations

For an algebraic variety $X$ over $\mathbb{C}$, we denote $H_k(X)$ as its Borel-Moore homology of degree $k$. Let us define the Poincare polynomial associated it by $$P(X)=\sum_{k\in \mathbb{N}}dim ...
1
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0answers
92 views

Cohomology of pushforward of coherent sheaf and connected components

Let $f: X\rightarrow Y$ be a proper morphism, $\mathscr F$ a coherent sheaf on $X$. If we know something about the connected components of fibers of $f$ can that give us a useful upper bound for $i$ ...
14
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3answers
512 views

Height functions on $\mathcal{M}_g(\overline{\Bbb{Q}})$ defined via dessins d'enfants?

Belyi's theorem establishes a correspondence between smooth projective curves defined over number fields and the so called dessins d'enfants which are bipartite graphs embedded on an oriented surface ...
2
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1answer
191 views

Quotient variety and subgroups

Let $G$ be an affine algebraic group (let's say over $\mathbb{C}$). If necessary one can assume $G$ to be reductive. Imagine one has $X$ over which $G$ acts freely: moreover, we have a locally closed ...
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0answers
74 views

Mapping points 1:1 on a surface

Again, see How can I "see" that a map is birational? — but I formulate my hypothesis generically. Let (the implicit surface) $F[x,y,z]$ be a polynomial with degree $2$ in all variables, i.e. ...
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0answers
110 views

understanding higher direct images of $\mathbb{G}_m$ for a finite Galois map

Let $X$ be a smooth quasi-projective variety over $\mathbb{C}$, and let $\mu_r$ denote the group of $r$-th roots of unity, and moreover suppose $\mu_r$ (algebraically) acts on $X$ freely. Then $Y:= X/\...
4
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0answers
174 views

Formality of a category of constructible sheaves

Let $X= S^1 \wedge S^1$ be a wedge of circles. Then $X$ admits a natural stratification $\mathcal{S}$ as a union of two disjoint open intervals $I_1, I_2$ and a point $\{*\}$. Let $D_{\mathcal{S}}(X)$ ...
4
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0answers
160 views

Stratified fibration property of the "Ran" affine Grassmannian

Let us consider the so-called Ran Grassmannian $Gr_{Ran}$, i.e. the geometric object defined e.g. in [Zhu, An Introduction to the affine Grassmannian and the Geometric Satake equivalence, Definition 3....
4
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0answers
336 views

Deligne's letter to Soulé from 1985

There is a famous letter of Deligne to C. Soulé in which, apparently, Deligne first formulated the conjecture on the existence of an abelian category of mixed motives, extending Grothendieck's pure ...
2
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1answer
264 views

Purity of perverse cohomology sheaves

Let $f\colon X\to Y$ be a morphism of projective varieties over a finite field. Let $K$ be a perverse pure sheaf on $X$. Are the perverse cohomology sheaves of $f_*(K)$ pure? I am just learning the ...
2
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0answers
152 views

Automorphism group in finite dimensional case

Let $K$ be a field, $G_a := (K, +)$ be the additive group of $K$, and $X$ an affine variety. I found the following claim: if $X$ admits a non-trivial $G_a$-action and $\dim(X) \ge 2$, then the group $\...
39
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0answers
1k views

Thomason's "open letter" to the mathematical community

In the 1989, Bob Thomason left his CNRS position in Orsay and moved to Paris VII. It was during this period that he composed his "Open Letter" to the mathematical community. The letter ...
5
votes
1answer
358 views

Why does $p_*p^! A$ deserve to be called homology with coefficients in $A$?

Let $p:X\to S$ be the unique map from a (locally compact) topological space $X$ to a point. Since $\underline{\hom}(\underline{\mathbb{Z}},-)$ is the identity functor, we have that $\Gamma(X,-)=\hom(\...
5
votes
2answers
355 views

Holomorphic retraction $\implies$ holomorphic tubular neighbourhood?

Let $M$ be a complex manifold and $S \subset M$ a compact complex submanifold together with a holomorphic retraction $$r : M \to S,$$ i.e. a holomorphic map which restricts to the identity on $S$. ...
4
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0answers
134 views

Over which fields does every algebraic curve of genus one have a rational point?

Over which fields does every smooth projective geometrically connected curve of genus one have a rational point?
4
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0answers
166 views

Is equation $y^3+x y + x^4 + 4 = 0$ solvable locally (in ${\mathbb Q}_p$ for all $p$)?

When finding out whether an equation in 2 variables has rational solutions (or, equivalently, whether an algebraic curve has any rational points), many authors recommend checking the local solubility ...
6
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1answer
235 views

Sets of $\mathbb{F}_p$-points of closed subvarieties of $\mathbb{A}^n$

Let $p$ be a prime and let $n\geq 2$ be an integer. The set $\mathbb{A}^n(\mathbb{F}_p)$ has $p^n$ elements so it has $2^{p^n}$ subsets. How many of those subsets are of the form $V(\mathbb{F}_p)$ ...
6
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1answer
244 views

Why does the Chern character descend to the numerical Grothendieck group for surfaces?

Let $X$ be a complex smooth projective variety of dimension $d$. Let $K(X) := K(\text{Coh}(X))$ denote the Grothendieck group of coherent sheaves on $X$. For two coherent sheaves $E$ and $F$ on $X$, ...
2
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0answers
240 views

How to deduce Künneth from its relative version (in cohomology of sheaves)

Let $p:X\to S$ and $q:Y\to S$ be morphisms of "spaces" over $S$. We have an isomorphism $$f_!(M\boxtimes N)=p_! M\otimes q_!N$$ in the derived category of "sheaves" over $S$, where ...
0
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0answers
86 views

The dimension of algebraic set $\{ F(x_1y_1, ..., x_my_m) = 0 \}$ compared to the dimension of $\{ F(x_1, ..., x_m) = 0 \}$

Let $F_i$ be homogeneous forms with $\mathbb{C}$ coefficients in $n$ variables for each $i$. Let $$ T_2 = \{ (\mathbf{x}, \mathbf{y}) \in \mathbb{A}_{\mathbb{C}}^{2n} : F_i(x_1y_1, ..., x_n y_n) =0, ...
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0answers
56 views

Non-torsion infinitely divisible elements in the Chow group

It was shown in "Clemens, Herbert, Homological equivalence, modulo algebraic equivalence, is not finitely generated.", that the Chow group mod algebraic equivalence of smooth complex ...
2
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0answers
195 views

For a nilpotent matrix A, are the cardinalities of sets: 1) B: commute with A, 2) B: anticommute with A, 3) B: q-commute with A — the same?

Let us work over finite fields $F_{p^k}$. Simulations seems to indicate: Question 1: Consider a nilpotent matrix $A$, consider the set of all matrices $B$, such that $AB-qBA=0$, then cardinality of ...
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0answers
135 views

Rational normal curve as determinantal variety

We work here over complex numbers. Let $\Omega(Z)$ \begin{pmatrix} L_1 & L_2 & ... & L_n \\ M_1 & M_2 & ... & M_n\\ \end{pmatrix} be a $2 \times n$ matrix of homogeneous ...
8
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0answers
319 views

Kähler metric on the Hilbert scheme of points on a surface

Question. Let $S$ be a non-singular complex projective surface and let $S^{[n]}$ be its Hilbert scheme of $n$ points. Is there a natural way to associate to a Kähler metric $\omega$ on $S$ a Kähler ...
3
votes
1answer
127 views

K-projectivity for rings of finite homological dimension

Let $R$ be a Noetherian commutative ring. A complex of $R$-modules $P^{\bullet}$ is K-projective if for any acyclic complex $A^{\bullet}$, the complex of abelian groups $ Hom(P^{\bullet}, A^{\bullet})$...
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0answers
162 views

Does Lemma 5.4 in Deligne's Ramanujan paper generalize to Shimura varieties of PEL type?

It is generally not known if a smooth variety over a perfect field embeds into a smooth proper variety. Lemma 5.4 in Formes modulaires et représentations $\ell$-adiques provides such an embedding for ...
5
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0answers
92 views

Parameter spaces for conic bundles

A conic bundle over $\mathbb{P}^n$ is a morphism $\pi:X\rightarrow\mathbb{P}^n$ with fibers isomorphic to plane conics. A conic bundle $\pi:X\rightarrow\mathbb{P}^n$ is minimal if it has relative ...
5
votes
0answers
308 views

Cohomology theories for algebraic varieties over number fields

There is a standard line which is repeated by anyone writing/talking about motives and cohomology of algebraic varieties over number fields: namely, there are many such cohomologies and then the ...
9
votes
2answers
568 views

Character variety of the free group

A classical result of Fricke--Klein--Vogt from the late 1800s implies that the character variety $\chi_\mathbb{C}$ associated to the free group $F_2$ and the algebraic group $\mathrm{SL}_2(\mathbb{C})$...
5
votes
1answer
275 views

Number of points of parabolic Springer fibres

Let $P$ be a parabolic subgroup of $\mathrm{GL}_n$ and $u\in P$ a unipotent element. The parabolic Springer fibre associated to $(P,u)$ can be defined by $$ \mathcal{P}_u:=\{gP\in G/P \mathrel\vert g^{...

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