# Questions tagged [ag.algebraic-geometry]

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

21,234
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### Finiteness of theta vanishing in the KP direction for locally planar curves

I believe the main question is Question 2 at the end, and for experts it might be completely okay to skip directly to it (assuming I'm not saying any nonsense).
My motivation comes from pure algebraic ...

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### Why the following quasi isomorphism implies the morphism to be a resolution (a step in the paper A characterization of rational singularities)

I was reading the paper A Characterization of Rational Singularities by Professor Kovács.
The main theorem is stated as follows:
THEOREM 1. Let $\phi: Y \rightarrow X$ be a morphism of varieties over ...

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### Unirational algebraic group scheme smooth

Let $G$ an unirational algebraic $k$-group scheme in sense of this book on Neron models, ie separated $k$-group scheme of finite type which is geometrically reduced & connected). Note we assume ...

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### Are covering families of localizations stable under pushouts?

For a commutative ring $A$, we call a finite family of localizations $A \to A_{S_i}$ (where $S_i$ are some subsets of $A$) a covering if the canonical morphism $A \to \prod A_{S_i}$ is an effective ...

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### The indecomposable bundle on an elliptic curve

M. Atiyah (Theorem 5, p. 432 of "Vector bundles on an elliptic curve") defines an indecomposable bundle of degree $0$ that has a global section for each rank $r$ (I'm thinking on an elliptic ...

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### Deform a divisor from a fiber in a fibration

Suppose $X\rightarrow Z$ is a projective smooth morphism. Let $0\in Z$ be a closed point, $X_0$ the corresponding fiber. Suppose $H^1(X_0,\mathcal{O})=H^2(X_0,\mathcal{O})=0$, then a line bundle $L$ ...

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### Is the weight-monodromy conjecture known for unramified representations?

Let $X$ be a smooth proper variety over a number field $K$, $v$ a place of $K$ lying over a prime number $p \neq \ell$, and $V := H^n(X_{\overline{K}};\mathbb{Q}_{\ell})$. Suppose $V$ is unramified at ...

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### A Weierstrass product theorem for invertible formal Laurent series over local Artinian rings?

Let $(A,\mathfrak{m},\kappa)$ denote a commutative local Artinian ring. Somewhat by accident, I've stumbled across the following interesting decomposition:
$$
A(\!(t)\!)^\times = t^\mathbb{Z} \cdot (1 ...

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141
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### The intersection number $C\cdot D=\deg(D_{/C})$

Let $S$ be an algebraic complex surface, and $D=[(U_\alpha,f_{\alpha})]$ is a Cartier divisor over $S$, and let $\cal{O}_S(D)$ be the sheaf associated to $D$. And let $C$ be a complex compact curve in ...

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### What is Pic of the torus global affine Grassmannian?

Let $T$ be a torus and $X$ a proper smooth curve over characteristic $0$ algebraically closed field $k$.
What is $\text{Pic}(\text{Gr}_{T,X^n})$?
Here $\text{Gr}_{T,X^n}$ is the BD Grassmannian over $...

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### Toric varieties as hypersurfaces of degree (1, ..., 1) in a product of projective spaces

I wonder if some hypersurfaces of multi-degree $(1, ..., 1)$ in a product of projective spaces are toric, with polytope a union of polytopes of toric divisors of the ambient space. This question is ...

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### Line bundles on curve with nodal singularity

Let $C$ be be an irreducible reduced curve over alg closed field $k$ with only one single nodal singularity $x$ and $f:N \to C$ it's normalization with $f^{-1}(x)=\{x_1,x_2\}$ (as set), and an iso ...

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### What is the meaning of universal family of Fulton Macpherson configuration space?

Fulton and Macpherson suggests the way to compactify the set of $n$-labelled distinct point on variety in their paper, "A Compactification of Configuration Spaces"
In this paper, the process ...

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### Multiplicities of components of a Springer fibre

Given a Springer fibre of type A, are multiplicities of its irreducible components known in general, or at least in the special cases of two-row/hook types?
By multiplicities I mean considering a ...

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### Nagata's compactification

Let $f:X \to Y$ be a separated and finite type morphism of schemes. We assume that there is an open immersion $i:U \to Y$ and a proper morphism $f_U:X \to U$ such that $f$ and $i \circ f_U $ coincide. ...

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### Blowing up and product

Let $X$ be a noetherian scheme over an algebraically closed field, $Y\subset X$ a closed subscheme, and $\widetilde{X}\to X$ the blowing up of $X$ along $Y$. Let $Z$ be a noetherian scheme. Is it true ...

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### Stable curves over non-noetherian schemes

In their seminal paper The irreducibility of the space of curves of a given genus, Deligne and Mumford define a stable curve of genus $g$ over a scheme $S$ to be a flat, proper morphism $X\to S$, all ...

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### Faithful flatness and non-commutative algebras

$\DeclareMathOperator\Spec{Spec}$When dealing with commutative algebras, a usefull criterion for faithful flatness is the following:
Let $f:A\rightarrow B$ be a morphism of commutative algebras. Then $...

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253
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### The Krull dimension of the tensor product of rings

The Krull dimension of a ring $R$ is defined as the length of the longest chain of prime ideals in it. Let $R_i$, for $i\in\mathbb{N}$ denote a sequence of commutative Noetherian rings of Krull ...

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### MMP for surfaces over a curve where the geometric generic fiber is a rational curve

I am looking for an explaination or an reference for the following fact:
Let $\pi:X\rightarrow Z$ be a contraction from a smooth surface $X$ to a curve $Z$. Assume that the geometric generic fiber of ...

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### Do the nearby cycle and Beilinson's vanishing cycle functors commute?

Let $X$ be a complex algebraic variety with a pair of regular functions $f_1,f_2$. To these functions we can associate various functors: the nearby cycles functor $\Psi_{f_i}$, the vanishing cycles ...

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### Existence of a hyper plane

I am very new to algebraic geometry, and self-studying varieties. I have the following question.
Suppose $Y$ is a variety of dimension $r$ and degree $d>1$ in $\mathbb{P}^n$. Let $P$ be a ...

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### Deformations over $A_{\inf}$

Setup:
Let $K$ be a perfectoid field of characteristic $0$ with tilt $K^{\flat}$.
Let $A_{\inf}=W(\mathcal{O}_{K^{\flat}})$ be the infinitesimal period ring.
Let $\mathcal{X}$ be a flat, projective $\...

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### Is the BGQ spectral sequence functorial with respect to morphisms of finite Tor-dimension?

It is well known that the BGQ (Brown-Gersten-Quillen) spectral sequence for the G-theory of a Noetherian scheme of finite Krull-dimension is contravariant with respect to flat morphisms.
My question ...

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### There exists noncommutative geometric invariant theory?

In this question, I am going to consider noncommutative projective algebraic geometry, as introduced by Artin and Zhang in the seminal paper Noncommutative projective schemes. The $\operatorname{Proj}$...

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### Normalizer of solvable linear group is an algebraic group?

I am trying to read the article "Three-dimensional affine crystallographic groups" of Fried–Goldman (Adv. Math., 1983). At some place, it states that if $G$ is a connected solvable closed ...

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### Grothendieck purity for Brauer groups of stacks

Let $X$ be a smooth variety over a field $k$ (for the sake of simplicity of characteristic $0$) and $\operatorname{Br}(X) := H^2_{\text{ét}}(X, \mathbb{G}_m)$ its (cohomological) Brauer group (...

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### Locus where a family of cycles is rationally trivial

Let $B$ be a smooth quasi-projective variety over a field of characteristic zero.
Let $\pi\colon \mathcal{X} \rightarrow B$ be a smooth and projective morphism with geometrically integral fibres. Let $...

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### Question regarding the definition of linearization of line bundles

I'm reading Dolgachev's book 'Lectures on invariant theory'. In Chapter 7, the linearization of a group action is discussed. Let $G$ be a linear algebraic group acting on a quasi-projective variety $X$...

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### Frobenius and regular scheme

Let $X$ be a noetherian regular scheme over $\mathbb{F}_{p}$. Then by Kunz's theorem, the absolute Frobenius $F: X\rightarrow X$ is flat and integral. Can it be written as a projective limit of finite ...

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### Infinitesimal criteria for unramified morphism on stacks

In an Artin stack, the tangent complex is fundamentally a derived object, often viewed as a 2-term complex via some quotient presentation, as discussed in Raskin's note here, for example. In ...

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### How to conclude the quasi-projective case of the derived McKay correspondence from the projective case?

I am currently trying to understand the paper "Mukai implies McKay" from Bridgeland, King and Reid (cf. here). Let me sum up the setting we find ourselves in:
Let $M$ be a smooth quasi-...

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### Are there geometric $\mathbb{G}_a$-quotients with trivial stabilizers, not being principal bundles?

Consider algebraic $\mathbb{C}$-schemes. The group scheme $\mathbb{G}_a$ is the scheme $\mathbb{A}^1$ with the addition. This is not a reductive group. Here I want to know some examples of $\mathbb{G}...

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### Reference for application of local cohomology to complex manifolds with points removed

Reference request - I am looking at Dolbeault cohomology on compact complex manifolds (not Riemann surfaces) with points removed. I have been told that the key to doing this is to look at Local ...

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### Morphisms of affine (embedded) varieties = morphisms as quasi-projective varieties? [migrated]

The following question occurred to me while reading An invitation to algebraic geometry by Smith (2000). The definitions are as in the book (and not up to debate). Everything is over $\mathbb{C}$ here....

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### Trace morphism for projective morphism on differentials forms

Let $k$ be a field, $X$ and $Y$ two connected $k$-varieties, and $f:X\rightarrow Y$ a dominant projective morphism of relative dimension $d$.
I would like to know under which condition there is a ...

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### Confusion regarding the invariant rational functions

I was reading S. Mukai's- An Introduction to Invariants and Moduli, and I came across Proposition 6.16 on page 195 (see the screenshot below)
It says that "every invariant rational function can ...

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### Three-dimensional analogues of Hirzebruch surfaces

There are several ways of describing a Hirzebruch surface, for example as the blow-up of $\mathbb{P}^2$ at one point or as $\mathbb{P}(\mathcal{O}_{\mathbb{P}^1} \oplus \mathcal{O}_{\mathbb{P}^1}(n))$....

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### Zero loci of sections of wedge product of bundles

Let $V$ be a $\mathbf{C}$-vector space of dimension $n$, and consider the Grassmannian $G:=Gr(2, V)$ of 2-dim subspaces of $V$. Then we have the tautological subbundle $E\subset V\otimes \mathcal{O}_G$...

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### Conjecture: Given any five points, we can always draw a pair of non-intersecting circles whose diameter endpoints are four of those points

The following question resisted attacks at Math SE, so I thought I would try posting it here.
Is the following conjecture true or false:
Given any five coplanar points, we can always draw at least ...

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### Adjunction correspondence for Blow up of double point

Let $C$ a curve over an algebr closed field $k$ with a singular double point singularity at $x$ and $\pi: C' \to C$ the blowup in $x$ and let $x_1,x_2 \in C'$ be the two points over $x$.
Why holds for ...

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### Calculate genus of reducible nodal curve

Let $C$ be be a connected reducible nodal curve over alg closed field $k$, such that all (finitely many) irred components $C_i$ of $C$ are smooth and intersections between different components are ...

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### Does every faithful action on a scheme act freely on a dense open subset?

Disclaimer: I have asked this question on math exchange a week ago (here), but sadly to no avail. So I decided to escalate my question:
Let $G$ be a finite group acting faithfully on a smooth quasi-...

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### Derived algebraic geometry and Bridgeland stability conditions [closed]

In the context of derived algebraic geometry can someone elucidate the intricacies of Bridgeland stability conditions on derived categories of coherent sheaves? Where can one find examples involving ...

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### Embedding of the first Hirzebruch surface in $\mathbb{P}^4$ as a cubic surface

The first Hirzebruch surface (the blow-up of $\mathbb{P}^2$ at one point) is a projective toric surface that naturally embeds into $\mathbb{P}^4$ as a cubic surface (sometimes called the cubic scroll)....

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### Anisotropic kernel of groups of type A

I'm studying the results of classification of reductive groups using Tits index and anisotropic kernel.
It is known that simple groups with Tits index $^1 A_{n,r}^{(d)}$ are of the form $SL_{r+1}(D)$, ...

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### Arbitrary base change of a parahoric subgroup in split case

Assume $R\subset R'$ are henselien discretly valued rings with fraction field $K$ and $K'$, $G$ is a semisimple split group over $K$. Consider the parahoric group scheme $\mathcal{P}_F$ over $R$ ...

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### Proof of a result by Zhang in Artin's seminal paper

In his seminal paper, Some open problems on three-dimensional graded domains, M. Artin proposed a very small list of possible division rings of fractions that can appear as 'noncommutative function ...

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### Smoothness of Picard scheme when $H^2(\mathcal{O}_{X_s})$ on fibers vanish

A question about the proof of Proposition 5.19 in Kleiman's notes on Picard scheme. Let $X$ be a $S$-scheme. Then the claim is that:
Assume that Picard scheme $\operatorname{Pic}_{X/S}$ exists and ...

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### $\mathbb{Z}[T]$-Solidification in light condensed setting

In the lectures to "Analytic Stacks" Scholze and Clausen introduced a new concept of "light" condensed mathematics. In Lecture 7 Clausen introduces the derived $T$-solidification ...