Questions tagged [ag.algebraic-geometry]

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

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2
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0answers
59 views

Tensor product by the canonical module preserves Cohen-Macaulayness

Let $X$ be a $\mathbb{Q}$-Gorenstein variety of dimension at least $2$. Suppose that $X$ is normal and Cohen-Macaulay with at worst isolated singularities. Let $F$ be a maximal Cohen-Macaulay $\...
5
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1answer
161 views

Intersection cycle in a product of Grassmannians

Let $G(k,n)$ denote the Grassmiannian of $k$-planes in $\mathbb C^n$. Let's define $$ I_j =\{ (\Lambda_1,\Lambda_2 ) \in G(k,n) \times G(l,n) \, | \, \dim(\Lambda_1 \cap \Lambda_2) \geq j \}. $$ These ...
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1answer
57 views

What is the pull-back of a polarization of abelian schemes over different bases?

The following came up when reading the definition of the moduli stack of principally polarized abelian varieties in [1]. Let $\pi_1:A_1 \to S_1$ and $\pi_2: A_2 \to S_2$ be abelian schemes over $S_i$, ...
2
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1answer
61 views

Principal bundles from a fibration of homogeneous spaces

Let $G$ be a compact (Lie) group, and $H \subseteq H'$ two compact (Lie) subgroups. It is clear that we have an obvious surjective map of homogeneous spaces $$ G/H \twoheadrightarrow G/H'. $$ Will it ...
4
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1answer
124 views

Curve with a rational point but no new points in number fields of low degree

Given an integer $d\geq 2$ is there an algebraic curve $C/\mathbb{Q}$ with $C(\mathbb{Q})\neq\emptyset$ and the natural map $C(\mathbb{Q})\to C(F)$ bijective for all number fields of degree at most $d$...
3
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0answers
132 views

Recovering classical Tannakian reconstruction from Lurie

$\DeclareMathOperator\QC{QC}$Let $k$ be a field and $G$ a smooth affine group scheme $k$. It is a "classical" fact that you can recover $G$ from the tensor category $\text{Rep}_k(G)$ of ...
6
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167 views

The geometric "hands-on" vs. algebraic approach to nearby cycles

Feel free to skip to the question below; the following is just context and discussion: An interesting, but seemingly less used result in the theory of nearby cycles of constructible sheaves is (...
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1answer
119 views

Bound for multiplicities of closed points on scheme

Let $K$ be a perfect field, and let $f_1, \ldots, f_m \in K[X_1,\ldots,X_n]$ be polynomials. Consider the affine scheme $$X = \mathrm{Spec}(K[X_1,\ldots, X_n]/(f_1,\ldots,f_m))$$ and let $N = \dim(X)$....
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0answers
151 views

How to construct a morphism $f_*\hom(X,f^!Y)\to \hom(f_!X,Y)$

Let $\mathscr{C}$ and $\mathscr{D}$ be closed symmetric monoidal categories. We fix a strong symmetric monoidal functor $f^*:\mathscr{D}\to\mathscr{C}$ with a right adjoint $f_*:\mathscr{C}\to\mathscr{...
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0answers
93 views

Integral lattice in noncommutative Hodge theory

Associated to a $DG_{\mathbb{C}}$-category, $\mathcal{C}$, we have some Hodge theoretic data - $HH_{*}(\mathcal{C})$ plays the role of Hodge cohomology and $HP$ plays the role of de Rham cohomology. ...
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0answers
88 views

Quiver varieties associated to D_4

Let $Q=(I,\Omega)$ be the $D_4$ affine quiver. We choose as dimension vector $(2,1,1,1,1)$ (where $2$ is on the central vertex). As this dimension vector is indivisible, we can choose a generic $\...
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0answers
63 views

Varieties with the same number of $\mathbb{F}_p$-points but different traces of Frobenius in some degree

Are there smooth proper varieties $X$ and $Y$ over $\mathbb{F}_p$ of the same dimension such that $X(\mathbb{F}_p)=Y(\mathbb{F}_p)$ but Frobenius has different traces on $H^i(X, \mathbb{Q}_l)$ and $H^...
17
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1answer
337 views

Varieties with the same number of $\mathbb{F}_p$-points for all but finitely many primes

If two varieties over $\mathbb{Q}$ have the same number of $\mathbb{F}_p$-points for all but finitely many primes do they have the same number of $\mathbb{F}_{p^n}$-points for all $n>1$ and for all ...
3
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0answers
127 views

In Deligne-Lusztig theory which degrees do irreps show up in?

In Deligne-Lusztig theory we take an alternating sum over cohomology in all degrees. Given an irrep of a finite group of Lie type can we trace back which degree it shows up in?
5
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107 views

Realize the discrete series of $\mathrm{SL}_2(\mathbb{F}_q)$ in an abelian variety

Is there an abelian variety $A/\mathbb{F}_q$ and an embedding $\mathrm{SL}_2(\mathbb{F}_q)\to \mathrm{Aut}_{\mathbb{F}_q}(A)$ such that $H^1(A\otimes \overline{\mathbb{F}_{q}}, \mathbb{Q}_l)$ contains ...
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0answers
46 views

Minimum level principal congruence subgroup coming from neat open compact subgroup

For a connected reductive group $G/\mathbb{Q}$ what is known about the minimum level such that the respective principal congruence subgroup is the intersection of a neat open compact subgroup of $G(\...
2
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1answer
155 views

Fundamental groups of degree 2 covers of projective spaces

Does being a degree 2 cover of a projective space impose restrictions on the fundamental groups of non-singular complex projective varieties? For curves it does not.
3
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2answers
164 views

Stable $\infty$-categories of derived equivalent varieties

When two varieties have equivalent derived categories of coherent sheaves are the stable $\infty$-categories of coherent sheaves also equivalent? Are the stable $\infty$-categories of varieties "...
11
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3answers
644 views

Does there exist some $p(x) \in \mathbb{Q}[x]$, deg$(p) > 1$, which maps $\mathbb{Q}$ onto itself surjectively?

Clearly this is impossible for $p$ of even degree, and I imagine that Cardano’s formula quickly reveals it to be impossible in the cubic case, although I have not checked in detail. My guess is that ...
2
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0answers
127 views

Definition of $\mathcal{O}_{\mathcal{X}}$-modules over a stack $\mathcal{X}$

$\newcommand\Sch{\mathrm{Sch}}$For a stack $\mathcal{X}$ in $(\Sch/S)_{\textrm{ét}}$, there is a site $(\Sch/\mathcal{X})_{\textrm{ét}}$ whose objects are $(T,t)$, where $T$ is an étale $X$-scheme and ...
3
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0answers
81 views

Minimal $b_2$ in Sarkisov's construction

In the paper On the structure of conic bundles. Math. USSR, Izv., 120:355–390, 1982, Theorem 5.10, Sarkisov constructed the first example of non-rational, rationally connected $3$-fold $X$ with $H^{3}...
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0answers
130 views

Non-singular variety covered by pairwise disjoint singular subvarieties

Can you cover a non-singular algebraic variety by pairwise disjoint singular closed subvarieties? Varieties are over an algebraically closed field of characteristic other than $2$ and $3$. In ...
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0answers
43 views

Vector bundle associated to orthogonal flag

Let $V$ be a $(2n+1)$-dimensional complex vector space endowed with a non-degenerate symmetric bilinear form $q:V\times V\to \mathbb C$. Fix the notation: $$ OG(n-1,n,V):=\{W_{n-1}\subset W_n\subset ...
2
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0answers
66 views

Direct way to get the Hyodo-Kato Galois action

A group object $G$ in smooth proper rigid spaces has formal semistable model (possibly after a finite extension). Hyodo-Kato give the cohomology of $G$ a $(\phi, N)$-model structure which by Fontaine-...
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0answers
88 views

Finding an injective envelope containing another injective envelope

Let $R$ be a local principal ideal domain (PID) with only two prime ideals $0$ and $P$, and let $M$ be an $R$-module. Let for $r\in R$ and $m\in M$, $rm\not=0$. Now if $E(rm)$ is a fixed injective ...
1
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1answer
107 views

Conics on a cubic scroll

Let $i:\mathbb F\hookrightarrow\mathbb P^4$ be a cubic scroll i.e. $\mathbb F\simeq \mathbb P(\mathcal O_{\mathbb P^1}(1)\oplus\mathcal O_{\mathbb P^1}(2))\overset{n}{\rightarrow} \mathbb P^1$ with $\...
4
votes
1answer
220 views

Are the fibers of a surjective polynomial submersion $\mathbb{C}^n\to\mathbb{C}$ all homeomorphic?

Are the fibers of a surjective polynomial submersion $\mathbb{C}^n\to\mathbb{C}$ all homeomorphic?
8
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0answers
206 views

Does Borel fixed-point theorem hold for Deligne-Mumford stacks?

Let $X$ be a proper Deligne-Mumford stack over $\mathbb{C}$ with an action by a complex torus $T$. Let $X^T$ denote the fixed locus. Question: Is the following statement true? ...
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0answers
59 views

Is there a variety such that a regular model is known but not a semistable model?

Is there a smooth projective variety over a complete discretely valued field such that a regular model is known but no semistable model is known? Models after a finite possibly ramified extension also ...
2
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0answers
75 views

Base change of Hodge-Witt cohomology

Let $k$ be a perfect field of characteristic $p$, and $L$ be a finite extension of $k$. For a smooth projective variety $X$ defined over $k$, we denote the base change $X \times_k L$ by $X_L$. In this ...
3
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0answers
73 views

Perfect dg-modules under faithfully flat extension

Let $k \subseteq \bar{k}$ be an extension of fields (Orlov in the reference below seems to indicate the same thing will hold for faithfully flat maps but the case of fields is enough for me). On page ...
6
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1answer
252 views

Ideals of $F_2[x_1, x_2, \cdots, x_n]/(x_1^2, x_2^2, \cdots x_n^2)$

I am interested in the poset of all ideals of the local ring $$R_n = \mathbb{F}_2[x_1, x_2, \cdots, x_n]/(x_1^2, x_2^2, \cdots x_n^2).$$ $n=1$ is trivial. $n=2$ takes little work and it is shown below....
1
vote
1answer
117 views

Counting $\mathrm{mod}\:p$ solutions of Diophantine equation in two variables taking $O(p^2)$ time

Are there Diophantine equations in two variables such that counting solutions $\mathrm{mod}\:p$ requires $O(p^2)$ time? Geometrically this means we have to sort through a positive proportion of the ...
2
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0answers
140 views

Zariski's main theorem without inverse limits

Consider the statement A proper birational map of varieties with a normal target has connected fibers. Is there a proof of it that does not involve inverse limits at all?
1
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0answers
146 views

Cohomology of quotient stack

Let $X$ be an algebraic variety over $\mathbb{C}$ (the ground field is not important but this makes things easier I think) and $G$ an algebraic group acting over it. Let's say we know that there's a ...
6
votes
2answers
395 views

Can the theory of elliptic functions developed from purely geometric considerations?

I always had this question, but was unable to get a definitive answer to it. There is the theorem of division of the arc length of the lemniscate with ruler and compass. So I always wondered, is it ...
5
votes
1answer
117 views

Convex hull of a variety in real space

I am a physicist currently working on a question posed as part of an algebraic geometric description of a physical set: I did not find a question that is closely related to what I am searching for yet,...
4
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1answer
190 views

A question related to fiber bundle

Let $f:\mathbb{C}^3 \to \mathbb{C}$ be a morphism of varieties such that it is a smooth fiber bundle. Can I say that the fiber is $\mathbb{C}^2$?
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0answers
60 views

Proof of Artin-Rees \ Krull intersection motivated by universal property of blowup

I was very confused by the proof of Artin-Rees \ Krull intersection theorem when I was younger. Now that I learnt about blow up- I saw the Rees algebra again and I want to now gain a better ...
2
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0answers
155 views

Degree of a conjugacy class in an algebraic group?

Let $G$ be a classical group over a field $K$ ($G=\textrm{SL}_n, \textrm{SO}_n, \textrm{Sp}_n$). Consider $G$ as embedded within the affine space $M_n\sim \mathbb{A}^{n^2}$ of $n$-by-$n$ matrices. Let ...
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0answers
217 views
+150

Does the language of fibred categories gives the commutativity of the diagrams in Residues and Duality?

In Residues and Duality, R. Hartshorne and A. Grothendieck say that there are a plethora of compatibilities that need to be shown in order to have a six functor formalism. For example, if $f:X\to Y$, $...
3
votes
1answer
105 views

Variety with two different $\mathrm{mod}\:p$ fibers

Can a smooth projective variety over $\mathbb{Q}_p$ have two smooth projective models with non-isomorphic $\mathrm{mod}\:p$ fibers? Can the $\mathrm{mod}\:p$ fibers have different number of rational ...
2
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0answers
75 views

$\bigoplus_{k=0}^{\infty}H^n(X,I^k\mathcal{F})$ is a finitely-generated $\bigoplus_{k=0}^nI^k-$graded module

Does anyone know where I can find a proof of the following result ? Given a Noetherian ring $A$, a proper morphism of schemes $X\rightarrow \operatorname{Spec}A$, a coherent $O_X-$module $\mathcal{F}$ ...
2
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0answers
73 views

Mayer-Vietoris sequence from a bicartesian square of commutative rings

An article that I am reading quotes the following theorem (5.3 p.481, reformulated to focus on the commutative case) from Algebraic K-Theory by Hyman Bass: Let $\require{AMScd}$ \begin{CD} A @>p_2&...
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0answers
69 views

Global sections of a vector bundle over $OG(2,7)$

Let us work over $\mathbb C$, using the Grothendieck projectivization $\mathbb P():=Proj(Sym())$. Consider a $7$-dimensional vector space $V$ endowed with a symmetric non-degenerate bilinear form $q:V ...
5
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0answers
196 views

How can Siu prove the lower semi-continuity of plurigenera by extension theorem?

(crossposted from m.se) I am a novice in the field of complex geometry. The following question seems to be so simple that no one have explained it in related papers. In the famous paper---Siu, ...
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0answers
86 views

Number of points in number fields on curve of genus at least 2

A smooth projective curve over $\mathbb{Q}$ of genus at least 2 has finitely many $K$-points for any number field $K$. What functions from number fields to $\mathbb{N}$ come up as the number of points ...
0
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0answers
82 views

On some loci of rings

Let $(R, \mathfrak m)$ be a Noetherian local ring. Let P be a property of $R$. Set $$ P(R) =\{\mathfrak p \in Spec(R)\,\,\, |\,\,\, R_{\mathfrak p}\, \, \mbox{is } P\},$$ $$ nP(R) =\{\mathfrak p \in ...
2
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0answers
70 views

Tate module whose maximal semisimple subrepresentation is a line

An abelian variety over $\mathbb{Q}_p$ is cool if the maximal semisimple subrepresentation of its Tate module is a line. Are there cool abelian varieties of arbitrarily high dimension? What about the ...
2
votes
0answers
77 views

All Galois characters showing up in cohomology of one family of varieties

Fix a prime $p$. Can we find a smooth proper map $X\to Y$ of $\mathbb{Q}_p$-varieties such that any given representation $\mathrm{Gal}(\overline{\mathbb{Q}_p}/\mathbb{Q}_p)\to \mathrm{GL}_1(\mathbb{F}...

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