Questions tagged [ag.algebraic-geometry]

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

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reference request: good reduction equivalent to crystalline etale cohomology

Suppose $X$ is an abelian variety over a $p$-adic field $K$, and it's well known that $X$ has good reduction is equivalent to the etale cohomology of $X$ is crystalline, and $X$ has semistable ...
Richard's user avatar
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One question about Manetti surface

I am reading Ascher-Devleming-Liu's paper "Wall crossing for K-moduli spaces of plane curves" theorem 5.2 ADL19 and l have some confusions about the proof. Theorem 5.2 states that fixed a ...
RedLH's user avatar
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22 views

Projectivity of equivariant K-theory of toric variety

I'm looking at Vezzosi and Vistoli's paper: Higher algebraic K-theory for actions of diagonalizable groups. In Theorem 6.9, they prove that the $T$-equivariant K-theory of a smooth projective toric ...
onefishtwofish's user avatar
2 votes
0 answers
95 views

About pushforward of a sheaf of divisor

Let $X$ be a normal variety over an algebraically closed field of arbitrary characteristic, $f:X'\to X$ a log resolution, $L$ a Cartier divisor on $X$, and suppose $L\sim_{\mathbb{Q},f}E$, where $E$ ...
nariri's user avatar
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56 views

Mirror of a local K3 surface

Is there any description of a mirror manifold of a (non-compact) Calabi-Yau threefold given by the total space of the trivial line bundle on a K3 surface? If yes, in what way is it a mirror? Thanks ...
Cranium Clamp's user avatar
2 votes
0 answers
51 views

Projective resolution of a quiver with relations

How do we compute the projective resolution of a representation of a quiver with relations. For example consider the Beilinson quiver $B_4$ $. with the relations ­$\{\alpha_j^k\alpha_i^{k-1}=\alpha_i^...
user52991's user avatar
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What are the Hodge and log Hodge groups of $M_{g,n}$?

I would like to know, ideally with a reference, what the Hodge and log Hodge numbers of the moduli space of stable curves $\bar M_{g, n}$ are. At the very least I'd like to know the genus zero case $g ...
Leo Herr's user avatar
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146 views

Lifting of quadrics containing a curve

Let $C \subset \mathbb{P}^r$ be a projective curve (over $k=\mathbb{C}$), smooth, irreducible and nondegenerate of degree $d$, ie the embedding line bundle $\mathcal{O}_C(1)=(\mathcal{O}_{\mathbb{P}^r}...
user267839's user avatar
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1 vote
1 answer
171 views

Shrinking the base field of an affine variety

This is a question on algebraic geometry/commutative algebra. Let $K,L$ be fields of characteristics zero and let $K\subset L$ be a field extension (I am interested in the case when this is ...
S.J.'s user avatar
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103 views

How to find a single-variable polynomial in a zero-dimensional ideal?

Given finitely many multivariate polynomials with algebraic coefficients that generate a zero-dimensional ideal, is there an easy way to find a nonzero single-variable polynomial in this ideal? If we ...
Dustin G. Mixon's user avatar
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Find stratification to decompose constructible sheaf to constant parts (example from Wikipedia)

I have a question about techniques used in determining the stratification over which a constructible sheaf falls into even constant pieces demonstrated on this example from Wikipedia. Let $f:X = \text{...
user267839's user avatar
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Symmetric 0-dimensional schemes with generic Hilbert function and Grassmannians

I've came across this problem while thinking about some properties of fat schemes. Let me give you an explicit (motivating) example: We have $S=\mathbb{C}[x,y,z]$, the coordinate ring of $\mathbb{P}^2$...
gigi's user avatar
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1 vote
0 answers
74 views

étale, smooth, and unramified for product of schemes

The question comes from Liu's book. I already asked it on mathstack and I post here since I didn't get any answer at now (https://math.stackexchange.com/questions/4873827/%c3%a9tale-smooth-and-...
Analyse300's user avatar
5 votes
1 answer
275 views

Is there an English translation of Monique Hakim's thesis?

Monique Hakim's thesis, published in 1972 as Topos annelés et schémas relatifs, has been referenced on a multitude of occasions. But I struggle to find a translation into English, even an informal one....
xuq01's user avatar
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3 votes
1 answer
359 views

Why is this polynomial factorizable? [closed]

I met a curious problem on factorizing a homogenerous polynomial of degree 9. Problem: Show that the following polynomial can be divided by $(a_1+a_2+a_3)$: \begin{align} &\quad\left| \begin{array}...
LichenSDU's user avatar
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10 votes
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641 views

Relation between motives and geometric Langlands

When working over a number field (or a function field over a finite field), one predicts that the Langlands program is related to the theory of motives over this field. There are several ways I have ...
JustLikeNumberTheory's user avatar
2 votes
1 answer
336 views

Zero set of prime ideal

Let $k$ be a field of characteristic $0,$ but not necessarily algebraically closed. Let $P$ be a prime ideal of $k[X]$ (where $X$ is an $n$-tuple of variables), and $V := \{x \in K^n: \forall p \in P\ ...
Antongiulio Fornasiero's user avatar
2 votes
1 answer
94 views

Normality and integrality of schemes and splitting of map from structure sheaf to (derived)pushforward of structure sheaf along proper birational map

Let $R, S$ be commutative Noetherian rings such that $R$ is a subring of $S$. If $S$ is a normal domain, and there exists an $R$-linear map $\phi: S\to R$ whose restriction on $R$ is the identity map, ...
Snake Eyes's user avatar
2 votes
0 answers
99 views

Flag variety type Beilinson resolution

The Beilinson resolution is a locally free sheaves resolution for sheaf $\Delta_*\mathcal{O}_{\mathbb{P}}$,where $\Delta: \mathbb{P}\to \mathbb{P}\times\mathbb{P}$ is the diagonal embedding of ...
fool rabbit's user avatar
3 votes
1 answer
258 views

Riemann-Hilbert problem via quiver description

The moduli space of Fuchsian systems over $\mathbb{P}^1$ with prescribed adjoint orbits conditions at poles a.k.a. additive Deligne-Simpson problem can be presented under purely quiver description.The ...
TaiatLyu's user avatar
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0 answers
105 views

Some questions about $\ell$-adic monodromy

I'm stucking on the proof of the Lemma 3.12 of A p-adic analogue of Borel’s theorem. Here $\mathcal A_{g,\mathrm K}$ is just a shimura variety defined over $\mathbb Z_p$, and full level $\ell$ ...
Richard's user avatar
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When $\langle u,v,w \rangle$ is a maximal ideal in $\mathbb{C}[x,y]$?

Let $u,v,w \in \mathbb{C}[x,y]$ and let $\langle u,v,w \rangle$ be the ideal generated by $u,v,w$. It is known that for two elements the following result holds: $\langle u,v \rangle$ is a maximal ...
user237522's user avatar
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4 votes
1 answer
282 views

Residues and blow ups

On a 2-dimensional complex manifold consider two functions which are meromorphic with singularities along two divisors which meet at a point. There is a residue from these meromorphic functions (...
Edwin Beggs's user avatar
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2 votes
1 answer
135 views

One question about K-moduli space of smooth plane conic curves

I am reading Ascher-Devleming-Liu's paper "Wall crossing for K-moduli spaces of plane curves" example 4.5 (2) (b) ADL 19 and l have some confusions. From Li-Sun's paper "Conical Kähler-...
RedLH's user avatar
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3 votes
1 answer
141 views

Image, upto direct summands, of derived push-forward of resolution of singularities

Let $\mathcal C$ be a full subcategory (closed under isomorphism also) of an additive category $\mathcal A$. Then, $\text{add}(\mathcal C)$ is the full subcategory of $\mathcal A$ consisting of all ...
Alex's user avatar
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-1 votes
0 answers
77 views

Zero of gradient of a sum of squares of polynomial

Let $f_1,\ldots,f_n\in \mathbb{R}[x_1,\ldots,x_n]$ have non zero constant jacobian determinant. Then is it true that $f_1^2+\cdots+f_n^2$ has any critical point? Let me explain a motivation. If ...
George's user avatar
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11 votes
2 answers
709 views

What relationship is there between repeated roots of discriminants and orders of roots of the original polynomials?

Disclaimer: I asked this problem several days ago on MSE, I'm cross-posting it here. The title sounds like a high school problem, but (as a grad student not in algebra) it feels subtle/deep. ...
Harambe's user avatar
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Felix Schuller's proof of the tautness of rational singularities

I'm reading this paper written by Felix Schuller. https://docserv.uni-duesseldorf.de/servlets/DerivateServlet/Derivate-24686/singularities-bib.pdf Corollary 5.7 writes A rational double point is taut ...
George's user avatar
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5 votes
2 answers
461 views
+100

Algebra/Algebraic geometry in statistical mechanics

This is a soft question. I am currently studying statistical mechanics and I found this one by chance: Algebraic statistical mechanics And I also found some workshops on interactions between ...
FFjet's user avatar
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2 votes
1 answer
203 views

Is a finite morphism of Deligne-Mumford stacks proper?

The situation that I am in is the following. Let $\mathcal{X}$ be a smooth Deligne-Mumford stack over a field $k$. Let $X$ be a $k$-scheme together with a morphism $\pi;\mathcal{X}\rightarrow X$ (you ...
Hajime_Saito's user avatar
2 votes
1 answer
133 views

Isomorphic IC sheaves induced from different locally closed subvarieties

Let us work with varieties over $\mathbb{C}$ and $D^{b}_{c}(X)$ the bounded constructible derived category of sheaves of $\mathbb{Q}$ vector spaces. Say $X$ and $Y$ are smooth locally closed ...
arczn's user avatar
  • 23
0 votes
0 answers
91 views

Prime to $p$ monodromy of local system on rigid variety

Suppose $F$ is a finite extension of $\mathbb Q_p$, and $X$ is a rigid variety over $F$. I saw in proposition 3.7 of Oswal, Shankar, Zhu, and Patel - A $p$-adic analogue of Borel's theorem: "Let $...
Richard's user avatar
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1 vote
0 answers
114 views

A basis of the weight space in the semi-invariant ring corresponding to the weight $\langle(2,3,2),\cdot\rangle$

I'm trying to understand Example 10.11.1 on page 225 of the book "An introduction to quiver representations" by Harm Derksen and Jerzy Weyman (see the attached screenshot below) I want to ...
It'sMe's user avatar
  • 767
2 votes
1 answer
133 views

Non-torsion points of Tate curves

Let $E$ be a Tate curve over a $p$-adic field $K$. Then there exists $q \in K^*$ with the valuation $v(q)>0$ such that $E(\overline{K})= \overline{K}^*/\left< q \right>$. So it is easy to see ...
Desunkid's user avatar
  • 247
4 votes
2 answers
168 views

References for $K$-orbits in $G/B$

Let $G$ be a reductive group, $K$ a symmetric subgroup of $G$ (e.g., fixed point of an involution), and $B$ a Borel subgroup of $G$. Then it is well known that $G/B$ has finitely many $K$-orbits. ...
Hadi's user avatar
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2 votes
0 answers
109 views

What are the categories of IND and PRO schemes?

below is a mathexchange question with no answers so I drop it here. I have some difficulties to figure out what the category of IND-schemes and PRO-schemes are, in particualer the relations with ...
Marsault Chabat's user avatar
2 votes
1 answer
162 views

Finite étale cover of factorial ring

Let $A$ be a regular factorial ring. Consider $B=A[X]/(P)$ such that $B$ is finite étale over $A$. When do we have that $B$ is also factorial?
prochet's user avatar
  • 3,432
5 votes
1 answer
172 views

Cohomology and base change for the structure sheaf along a smooth proper morphism

Question: let $f : X \to S$ be a smooth proper morphism of schemes. Under what circumstances is it true that $R^i f_* \mathcal{O}_X$ is a locally-free sheaf whose formation commutes with all base ...
Ben C's user avatar
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5 votes
1 answer
204 views

Deformation invariance for tensor powers of the cotangent bundle

For a family of smooth projective varieties over the complex numbers $\mathbb{C}$, it's well-known that: The Hodge numbers $h^q(X,\Omega_X^p)$ are constant in the family (since they are ...
Irwin's user avatar
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0 votes
0 answers
49 views

Approximating open subset of profinite group by union of cosets of ideal

I am trying to understand the proof of Theorem 1.3 in this paper by poonen. Poonen refers to Lemma 20 in a different paper. He claims that the open subset $U_P \subseteq \hat{\mathcal{O}}_P$ can be ...
jb1403's user avatar
  • 1
3 votes
2 answers
221 views

Question about surface singularities

Throughout, $X$ will be a projective surface. I am looking for examples of the following surface singularities, I) A rational singularity that is not quotient. Obviously, it has to be non-Gorenstein, ...
Rio's user avatar
  • 231
1 vote
0 answers
81 views

Describing the hyperbolic structure of punctured torus in terms of the period lattice

Let $T$ be a torus, $T^* = T - \{p\}$ be the complement of a point $p$. Let's fix a pair of generators $x,y\in\pi_1(T^*)$. Their images in $\pi_1(T)$ also generate, and will also be denoted by $x,y$. ...
stupid_question_bot's user avatar
2 votes
0 answers
89 views

Singularities of curves over DVRs with non-reduced special fibre

Let $R$ be a complete DVR of mixed characteristic with fraction field $K$ of characteristic $0$ and residue field $k$ of characteristic $p>0$. Suppose that $\mathcal{X}$ is a normal $R$-curve such ...
David Hubbard's user avatar
1 vote
0 answers
61 views

Chow ring of simplicial toric varieties

Let $k$ be an algebraically closed field of characteristic zero. Let $X$ be a simplicial toric variety over $k$. In the 2011 book Toric Varieties by Cox, Little and Schenck, there is a theorem that ...
Boris's user avatar
  • 501
3 votes
1 answer
189 views

Original proof of Lefschetz's theorem on $(1,1)$ classes

Is there a "modern" account of Lefschetz proof of his theorem about $(1,1)$ classes for projective surfaces ? I believe that would be very interesting to understand the original arguments ...
Nicolas Hemelsoet's user avatar
-2 votes
0 answers
78 views

System of polynomial equations and its Jacobian determinant

double post https://math.stackexchange.com/questions/4875170/system-of-polynomial-equations-and-its-jacobian-determinant Does this propostion hold? Proposition Let $\mathbb{C}[x_1,...x_n]$ be a ...
George's user avatar
  • 449
4 votes
1 answer
147 views

Some questions on derived pull-back and push-forward functors of proper birational morphism of Noetherian quasi-separated schemes

Let $f: X \to Y$ be a proper birational morphism of Noetherian quasi-separated schemes. We have the derived pull-back $Lf^*: D(QCoh(Y))\to D(QCoh(X))$ (https://stacks.math.columbia.edu/tag/06YI) and ...
strat's user avatar
  • 291
10 votes
1 answer
362 views

Why can we take the colimit over the category of elements?

I'm trying to understand J. P. Murre's Tata notes on Grothendieck's theory of the fundamental group. For a Galois category $\mathcal C$ (which I'm taking to be locally small) with fundamental functor $...
themathandlanguagetutor's user avatar
1 vote
1 answer
100 views

Frobenius action on the trivial connection

Let $F$ denote the absolute Frobenius acting on a smooth quasiprojective scheme $X$ over a finite field $k$. Denote the trivial connection on $\mathcal{O}_X$ by $d$. Denote its pullback by Frobenius ...
kindasorta's user avatar
  • 1,373
0 votes
0 answers
33 views

Finding integral points of quadric without degree 1 terms

I consider for some $n\in\mathbb{N}$ the index set $I\subset\binom{n}{2}$ the following polynomial $p_I\in\mathcal{R}:=\mathbb{R}[x_1,...,x_n]$ with $$p_I(x_1,...,x_n)=\sum_{\lbrace i,j\rbrace \in I}(...
Jens Fischer's user avatar

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