# Questions tagged [ag.algebraic-geometry]

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

21,735
questions

7
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3
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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. ...

1
vote

1
answer

115
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### 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 ...

1
vote

0
answers

120
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### 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 $\...

0
votes

0
answers

93
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### 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 ...

5
votes

0
answers

121
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### 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 ...

4
votes

0
answers

167
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### 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-...

11
votes

1
answer

808
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### 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 ...

3
votes

1
answer

170
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### 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 ...

0
votes

0
answers

93
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### 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 ...

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 $H^1(JX,\mathbb{Z}_{\ell}(1))^{\vee}$ ...

0
votes

0
answers

52
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### 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 ...

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 $...

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) ...

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}...

-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 ...

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 $x\in X$ ...

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 \...

2
votes

1
answer

215
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### 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 ...

3
votes

1
answer

520
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### 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 $(x, y) \in \mathbb{S}^2 \times \mathbb{S}^2$, ...

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.
$(...

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 ...

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 ...

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 ...

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 ...

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 ...

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 ...

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{...

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 ...

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 ...

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*}
...

3
votes

0
answers

104
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### 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 ...

3
votes

0
answers

139
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### 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 ...

1
vote

0
answers

130
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### 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 ...

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 ...

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 ...

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 ...

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 ...

2
votes

2
answers

237
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### 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, ...

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}=...

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 $M_{1,1}$).
...

1
vote

1
answer

159
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### 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 ...

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 $g$ ...

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 ...

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) @&...

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?

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,...

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 ...

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 ...

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 ...