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Questions tagged [ag.algebraic-geometry]

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

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Non-noetherian schemes with noetherian underlying space (in the Zariski topology)

I am curious if there are many non-Noetherian schemes with Noetheiran underlying space. I know an example, we consider: $R= k[x_1, x_2, \dots]/\langle x_1^2, x_2^2, \dots\rangle $ for $k$ a field. ...
David Bowman's user avatar
1 vote
1 answer
115 views

Fitting closed polynomial curves to given points in the plane

Consider the equation $$ x^2 + y^2 - R^2 = 0 \tag{1}\label{470162_1} $$ which describes a circle. For a given point (e.g. $P_1 = (0.8, 1.5)$) the parameter $R$ can be easily determined. I am ...
cknoll's user avatar
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1 vote
0 answers
120 views

Is this formula for certain structure constants of quantum Schubert polynomials known?

Quantum Schubert polynomials $\mathfrak{S}_u^q(x)$ indexed by $S_\infty$ are polynomials in the polynomial ring $\mathbb{Z}[x,q]$ in infinitely many variables that form a basis of this ring over $\...
Matt Samuel's user avatar
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0 votes
0 answers
93 views

Higher direct images of locally constant etale sheaf under smooth proper map locally constant

Let $f:X \to Y$ a surjective smooth proper map between Noetherian schemes and $F$ a locally constant sheaf on small etale site of $X$. Question: Refering to Donu Arapura's answer here, how to see that ...
user267839's user avatar
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5 votes
0 answers
121 views

The proof of the fundamental theorem of tropical algebraic geometry in Maclagan-Sturmfels

I am trying to understand the proof of the fundamental theorem of tropical algebraic geometry from Maclagan-Sturmfels (Introduction to Tropical Geometry, Section 3.2 of the 2015 edition). Is the ...
pinaki's user avatar
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4 votes
0 answers
167 views

Relation between exotic sheaves in Achar's notes and in Bezrukavnikov-Mirkovic

I am trying to calculate explicitly a certain simple exotic sheaf (a simple object of the heart of the exotic t-structure on the Springer resolution, which is defined in Theorem 1.5.1 of Bezrukavnikov-...
Yellow Pig's user avatar
  • 2,540
11 votes
1 answer
808 views

Reference for a statement from Gaitsgory's thesis

In his PhD thesis, Gaitsgory (in his "remark 6") makes the following claim: Consider two complexes of holonomic $D$-modules with regular singularities on a variety $X.$ Suppose that at each ...
Michael Barz's user avatar
3 votes
1 answer
170 views

Is the cap-product map injective for singular varieties?

Let $X$ be a singular, projective (complex) variety of dimension $n$ with at worst isolated singularities. We know that taking cup-product with the cohomology class of $X$ induces a morphism from the ...
user45397's user avatar
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0 votes
0 answers
93 views

Principle of degeneration as precursor of Zariski's connectedness theorem (geometric intuition)

I have following question about so-called "principle of degeneration" in algebraic geometry (which in modern terms is an immediate consequence of Zariski's main theorem and goes in it's ...
user267839's user avatar
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7 votes
0 answers
137 views

Is the $\ell$-adic cohomology ring of a cubic threefold a complete invariant?

The only interesting $\ell$-adic cohomology of a smooth cubic threefold $X$ is $H^3(X,\mathbb{Z}_{\ell}(2))$, which is isomorphic as a $\mathrm{Gal}_k$-module to $H^1(JX,\mathbb{Z}_{\ell}(1))^{\vee}$ ...
TCiur's user avatar
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0 votes
0 answers
52 views

Combinatorial counting question related to count (anti)commuting N-tuples of matrices (more generally $(X_1,...X_n): F(X_i,X_j)=0$ - only one F)

Consider some finite set $S$ (can be matrices over $F_p$), consider some symmetric relation $F(s1,s2)$ which values are True or False (for example - matrices (anti)commutate or not). Question 1: can ...
Alexander Chervov's user avatar
2 votes
1 answer
71 views

Specialization of w-contractible objects on intersections on the pro-étale site

I'm trying to understand sections [61.25] and [61.26] of Stacks Project on closed immersions and extension by zero on the pro-étale site. Lemma [61.25.5] refers to affine weakly contractible objects $...
Absent mind's user avatar
4 votes
1 answer
235 views

Origin of the name Trace resp Integral symbol for the trace map of Dualizing Sheaf

Let $X \subset \mathbb{P}^n_k$ be a normal projective subscheme over $k$ of dimension $n$. The dualizing sheaf is in context of Serre duality a pair $(\omega_X,t)$ (which exists in that case) ...
user267839's user avatar
  • 6,064
2 votes
0 answers
137 views

Absolute Bloch-Kato Cohomology

The étale cohomology $R\Gamma_{\mathrm{ét}}(X;\mathbb{Z}_p(n))$ of a scheme $X/K$ can be computed by a Hochschild-Serre spectral sequence with terms of the form $H^i(K;H^j(X_{\overline{K}};\mathbb{Z}...
David Corwin's user avatar
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-1 votes
0 answers
86 views

Computing the Chow group of zero cycles on toric varieties

Let $X$ be a toric variety coming from a fan $\Delta$. It was mentioned in the 1997 paper Intersection Theory on Toric Varieties by Fulton and Sturmfels that the combinatorial presentation of Chow ...
Boris's user avatar
  • 549
2 votes
0 answers
82 views

Canonical model and the existence of general hyperplane

A variety is a separated reduced scheme of finite type over an algebraically closed field $k$ not necessarily affine. Let $X$ be a normal variety and $X$ has only one isolated singularity at $x\in X$ ...
George's user avatar
  • 199
4 votes
1 answer
214 views

Characterizing principal polarizations of abelian surfaces

Suppose $X$ is a complex abelian variety of dimension 2. Then I believe the ring of endomorphisms $\mathrm{End}(X)$, tensored with $\mathbb{C}$, is isomorphic to a subalgebra $M_2(\mathbb{C})$ of $2 \...
John Baez's user avatar
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2 votes
1 answer
215 views

Normalizer of Levi subgroup

Let $G$ be a reductive group (we can work on an algebraically closed field if needed) and let $L$ be a parabolic subgroup, i.e. the centralizer of a certain torus $T \subseteq G$. Associated with this ...
a_g's user avatar
  • 53
3 votes
1 answer
520 views

Zeros of a function defined on $\mathbb{S}^2 \times \mathbb{S}^2$

Let $u$ be a smooth function on the sphere, and for each $y \in \mathbb{S}^2$, let $R_y$ be the $180^\circ$ rotation about the vector $y$. For each pair $(x, y) \in \mathbb{S}^2 \times \mathbb{S}^2$, ...
MathLearner's user avatar
2 votes
1 answer
111 views

A property of canonical singularity

Let $X$ be a normal variety with only one singularity at $x$ and $(X,x)$ is a canonical singularity i.e. $(X.x)$ satisfies $(i)$ and $(ii)$. $(i)$ $(X,x)$ is a $\mathbb{Q}$ Goreinstein singularity. $(...
George's user avatar
  • 199
3 votes
2 answers
382 views

Guess the next polynoms in the sequence (MO vs. AI :), count anticommuting $F_p$-matrices, P. Hrubeš conjecture

Here is a sequence of polynoms - (presumably) counting N-tuples of ANTI-commuting 2x2 matrices over $F_p, p>2$. (That is just the case of 2x2 matrices, and (surprisingly) it is not so easy to see a ...
Alexander Chervov's user avatar
3 votes
0 answers
126 views

Computation of the Picard-Fuchs eq. and Gauss-Manin connection

I saw the classic example of the Picard-Fuchs eq. in almost any paper where the Gauss-Manin connection is mentioned in. The example is (Taken from Movasati-A Differential Introduction to Modular Forms ...
Or Shahar's user avatar
  • 451
4 votes
1 answer
216 views

On the bounded derived category of sheaves with coherent cohomology

Let $(X,\mathcal{O}_X)$ be a locally ringed space such that $\mathcal{O}_X$ is locally notherian, and let $\operatorname{Coh}(\mathcal{O}_X)$ be the category of coherent $\mathcal{O}_X$-modules. The ...
Fernando Peña Vázquez's user avatar
5 votes
1 answer
202 views

Closed complement of an open immersion of rigid analytic spaces

I am new to rigid analytic spaces (over non-archimedean fields) and I am confused about the notions of closed and open immersions. My question is are these two notions are "complement" of ...
Alexey Do's user avatar
  • 773
2 votes
0 answers
145 views

Unramified lisse $\overline{\mathbb{Q}}_{\ell}$-sheaves

Let $X$ be a connected noetherian scheme and $\ell$ a prime invertible on $X$. Let $D \subset X$ be a regular effective Cartier divisor (or more generally a normal crossings divisor, I suppose). Write ...
Hugo Zock's user avatar
1 vote
1 answer
269 views

Proj construction and nilpotent homogenous elements in graded ring

Let $A= \bigoplus_{n \ge 0} A_n$ be a commutative Noetherian graded ring and $f \in A_d$ a nonzero homogeneous element of degree $d>0$. The natural ring map $q:A \to A/(f)$ induces a well defined ...
user267839's user avatar
  • 6,064
11 votes
0 answers
232 views

Big list of Hochster dual concepts

Let $X$ be a spectral space. Then there is a canonical space $X^\vee$ with the same points, same constructible topology, and the opposite specialization order. This is known as “Hochster duality”, and ...
0 votes
2 answers
244 views

Vakil exercise on sheaf associated to the divisor of rational section

This is exercise 15.4.G. of Vakil's notes. Let $\mathscr{L}$ be an invertible sheaf on an irreducible normal scheme $X$ with $s$ a rational section of $\mathscr{L}$. We want that $\mathscr{O}_X(\text{...
Teddy's user avatar
  • 29
2 votes
0 answers
84 views

Reference request The support of $f$-nef divisor

I'm seaching for a proof of the theorem below. Do you know any reference? Consider $f:X\rightarrow Y$ projective birational map between normal varieties and $\mathbb{R}$ cartier divisor $D$ whose ...
George's user avatar
  • 199
2 votes
1 answer
126 views

Count N-tuples of commuting matrices over $F_q$ is given by polynomials with pattern $\sum q^{A_i(N)} P_{i}(q) $, where $P_i$ - do not depend on $N$?

Count pairs of $k \times k$ commuting matrices over finite field $F_q$ is given by certain polynomials in $q$ (which is quite rare phenomena for algebraic varieties) and have interesting generating ...
Alexander Chervov's user avatar
2 votes
2 answers
225 views

Finding rational points on intersection of quadrics in affine 3-space

Consider the subvariety of Spec $\mathbb{Q}[x,y,z]$ cut out by the equations \begin{eqnarray*} f_1: a_1x^2 - y^2 - b_1^2 & = & 0 \\ f_2 : a_2x^2 - z^2 - b_2^2 & = & 0 \end{eqnarray*} ...
stupid_question_bot's user avatar
3 votes
0 answers
104 views

Generalizing the Narasimhan–Seshadri theorem

There is a theorem that (stable, topologically trivial) holomorphic $G$-bundles are in one-to-one correspondence to flat $K$-bundles (with the appropriate corresponding condition), where $K$ is the ...
Charles Wang's user avatar
3 votes
0 answers
139 views

Examples of curves $C$ with $\operatorname{Jac}(C) \cong E^3$, $E$ a CM elliptic curve

Let $k$ be a field of your choice— I'm particularly interested in algebraically closed fields. Are there explicit examples of curves over $k$ whose Jacobian is isogenous to the product of three copies ...
Bma's user avatar
  • 301
1 vote
0 answers
130 views

The equation of cubic surface

I was studying the nature of singularities of cubic hypersurface on $\Bbb P^{n+1}$. The chosen form was $$f(x_0,x_1 \dots , x_{n+1})=x_0^3+x_1^3+\dotsb+x_{n+1}^3-c(x_0+x_1+\dotsb+x_{n+1} )^3=0.$$ I ...
mecid. s.'s user avatar
2 votes
1 answer
227 views

Sheaves which are locally free on subschemes of dimension zero

Let $\mathscr{F}$ be a coherent sheaf on a scheme $X$ with reasonable assumptions. Obviously if I restrict to any point $x \in X$, the restriction $\mathscr{F}|_x$ is free over $x$. I am interested in ...
ofiz's user avatar
  • 635
5 votes
1 answer
371 views

Finiteness of the Brauer group for a one-dimensional scheme that is proper over $\mathrm{Spec}(\mathbb{Z})$

Let $X$ be a scheme with $\dim(X)=1$ that is also proper over $\mathrm{Spec}(\mathbb{Z})$. In Milne's Etale Cohomology, he states that the finiteness of the Brauer group $\mathrm{Br}(X)$ follows from ...
user avatar
3 votes
1 answer
325 views

"Noetherianess" of $\mathrm{Mod}(\mathbb{F}_{p,\square})$

In classical commutative ring theory it is quite immediate to see, that a field is noetherian in the following sense: For any finitely generated $k$-vectorspace $M$, any sub-object is finitely ...
Jonas Heintze's user avatar
1 vote
0 answers
108 views

Monomorphism which is locally of finite presentation

$\DeclareMathOperator\Spec{Spec}$Let $X$ be an affine scheme of finite type over a field. Let $Y\to X$ be a monomorphism of schemes, which is locally of finite presentation. Is it true that $f$ is ...
user526421's user avatar
2 votes
2 answers
237 views

Artin's "Autoduality of the Jacobian"

In some of his papers (for example, in "Formal groups arising from algebraic varieties" with B. Mazur), M. Artin cites M. Artin and B. Wyman, Autoduality of the Jacobian, Bowdoin College, ...
Gabriel's user avatar
  • 1,049
2 votes
1 answer
197 views

Infinitesimal neighborhood and Ext group

$\DeclareMathOperator\Ext{Ext}$Let $X$ be a smooth projective complex variety and $\iota\colon Y\subset X$ be a smooth closed subvariety. It is well-known that there is a spectral sequence $$E_2^{p,q}=...
Jooh's user avatar
  • 141
3 votes
1 answer
243 views

Endomorphism ring of a generic elliptic curves in positive characteristic

Let E be a generic elliptic curve over an algebraically closed field $k$ of characteristic $p>0$ (i.e. an elliptic curve corresponding to a geometric point over the generic point of $M_{1,1}$). ...
aya 's user avatar
  • 177
1 vote
1 answer
159 views

Two different resolution of a three fold

Let $X\subset \mathbb{A}^4_\mathbb{C}$ be a three fold defined by the equation $xy-zw=0$ This variety has a singularity at origin of $\mathbb{A}^4_\mathbb{C}$ If we blow up this three fold in two ways ...
George's user avatar
  • 199
2 votes
1 answer
150 views

How to decompose a given polynomial by ideal generators

Given a finite set of polynomials $f_1, f_2,..., f_n$ of variables $x_1,...,x_m$, generating the ideal $I$, suppose that we have one more polynomial $g\in I$. What is the algorythm for decomposing $g$ ...
Dmitri Scheglov's user avatar
3 votes
1 answer
120 views

Is inverse image along finite group quotient $t$-exact for the perverse $t$-structure?

Let $q: \mathbb{C}\to \mathbb{C}$ be the quotient by $\mathbb{Z}/n\mathbb{Z},$ i.e. the map taking $z\mapsto z^n$. In the accepter answer to Operations on perverse sheaves on disk the inverse image of ...
Sergey Guminov's user avatar
5 votes
1 answer
169 views

A pushout diagram of derived categories coming from an open cover of schemes

Suppose $X=U\cup V$ is the standard open cover of $X=\mathbb{P}^1$ by two affine lines. The descent theorems say that the diagram (with all arrows restriction maps) $\require{AMScd}$ \begin{CD} D(X) @&...
Kaiyi Chen's user avatar
0 votes
0 answers
55 views

Reference for packing property and König property

Can someone please suggest reference material to study about the packing property and König property of ideals and some examples?
Sowbarnika R's user avatar
5 votes
2 answers
663 views

A version of Hilbert's Nullstellensatz for real zeros

$\newcommand\R{\Bbb R}$Let $Q(x_1,\dots,x_n)\in\R[x_1,\dots,x_n]$ be an irreducible polynomial such that the dimension of the set $Z:=\{(x_1,\dots,x_n)\in\R^n\colon Q(x_1,\dots,x_n)=0\}$ (defined, say,...
Iosif Pinelis's user avatar
0 votes
1 answer
204 views

Atiyah sequence of a coherent sheaf

I want to show that if $X$ is a smooth complex projective variety, then analytification induces an equivalence of categories between algebraic integrable connections on $X$ and analytic integrable ...
Tanny Sieben's user avatar
1 vote
0 answers
50 views

Leibniz formula for Fulton's divided difference operators for quantum Schubert polynomials

Schubert polynomials are polynomials in the ring $\mathbb{Z}[x]$ where $x$ is an infinite set of variables indexed by the positive integers and they can be expressed in terms of "standard ...
Matt Samuel's user avatar
  • 2,098
1 vote
0 answers
78 views

Vanishing of chow group of 0-cycles for affine, simplicial toric varieties

Let $k$ be an algebraically closed field of characteristic zero. Let $X$ be an affine, simplicial toric variety over $k$. If $X$ has dimension one, then it is the affine line over the field $k$, so ...
Boris's user avatar
  • 549