Questions tagged [ag.algebraic-geometry]

for questions on algebraic geometry, including algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology.

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76 views

Independent conditions imposed by a collection of double points

Let's consider the following statement : There exist a collection of $d$ points $\gamma \subset \mathbb{P}^{n}$, so that $h_{\Bbb P^n}(\gamma^{2} ,m) = \min\{(n+1)d, \binom {n+m}{n}\}$ implies for any ...
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195 views

Is the schematic Fargues--Fontaine curve universally closed over $\mathbb{Q}_p$? [closed]

I have been thinking about this question and it appears that the so-called Fargues--Fontaine curve may be relevant. A definition is given in this document A curve is a regular, Noetherian, ...
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1answer
272 views

Basic example of a formal affine scheme, functorial point of view

$\let\opn=\operatorname$For my BA thesis I have to describe formal groups from the functorial point of view. I am hence reading Strickland - Formal Schemes and Formal Groups, which is apparently the ...
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150 views

Coproduct in the category of affine schemes, functorial point of view

$\let\opn=\operatorname$An affine scheme is defined as a covariant representable functor $X:\opn{CRing} \to \opn{Set}$. The Yoneda embedding implies that the category of affine schemes, $\opn{...
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1answer
126 views

Factorize a morphism into a morphism locally of finite type and a quasi-compact morphism

Does there exist a scheme not admitting a morphism locally of finite type to a quasi-compact scheme? The reason I am asking this is that being locally of finite type and being quasi-compact are ...
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119 views

Universal closure of schemes à la Nagata

Nagata compactification theorem is the following fundamental result: Let $S$ be a qcqs scheme. Let $X$ be a separated $S$-scheme of finite type. Then there exists a proper $S$-scheme $\overline{X}$ ...
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166 views

Glue DVR to itself, get a separated non-affine scheme

Here is what seems to be a fun little exercise in algebraic geometry. Take a DVR $R$ and automorphism of $Frac(R)$; glue $Spec(R)$ to itself via this automorphism. Can the glued scheme be separated ...
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59 views

Defining pull-back of Chow groups under a morphism of special type

Let $X$ be a normal complex projective variety (not necessarily smooth), and let $Y$ be a smooth complex projective variety. Let $Z\subset X$ be a smooth closed subvariety. Let $\pi : Y\rightarrow X$...
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135 views

Brauer Group of a nodal curve

What is known about the Brauer Group of a Nodal curve (complete integral curve) over $k$ with singularity as ordinary double point? Is it trivial if $k$ algebraically closed?
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0answers
258 views

Morphisms of smooth varieties

Let $f:X\rightarrow Y$ be a surjective morphism of varieties (integral separated schemes of finite type over an algebraically closed field) such that $Y$ is smooth projective and all the fibers are ...
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1answer
139 views

Localization and containment in commutative ring

Let $R$ be a commutative ring with identity and $x, y $ be fixed elements of $R$ such that for each maximal ideal $m$ of $R$ we have $\langle \frac{x}{1_m}\rangle\subseteq\langle \frac{y}{1_m}\rangle$ ...
2
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0answers
111 views

Exact sequence of normal cones

Suppose that we have a sequence $i: X \hookrightarrow Y$, $j: Y \hookrightarrow Z$ of closed embeddings of varieties such that $i$ is regular. In this case, do we have an exact sequence of cones of ...
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271 views

A purely algebraic argument for existence of a section of a smooth projective morphism to the projective line

If I am reading this post correctly, any smooth projective $\mathbb{C}$-morphism of schemes $X\rightarrow \mathbb{P}^1$ admits a section. I am afraid of the topological argument presented there. Is ...
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96 views

Question about Local Henselian Rings

I have a question regarding properties/characterizations of local Henselian rings exploited in M. Artin's article "On Isolated Rational Singularities of Surfaces": Here the relevant excerpt: Remark: ...
5
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1answer
253 views

Reductive groups in algebraic geometry [duplicate]

In a lot of fields in algebraic geometry (e.g. GIT or topics on étale cohomology) which make use of group scheme concepts (or in more tame way of algebraic groups), the class of reductive algebraic ...
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418 views

Theorem from Deformation Theory

My question refers to some steps it the proof of Theorem 3.3 part (b) in Christensen's paper treating Deformation theory (see pages 9-11): https://mathematics.stanford.edu/wp.../A.-Christensen-Draft....
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49 views

Finding a cover of the Hilbert functor, that corresponds to a cover of the Grassmannian

Let $\mathrm{Grass}_{k+1,n+1}:\mathrm{Sch}^{\circ}\rightarrow\mathrm{Set}$ be the Grassmann functor, which maps a scheme $S$ to the set: $$\left\{\mathscr{U}\subseteq\mathscr{O}_S^{n+1}:\,\,\mathscr{...
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0answers
34 views

Pullback of homogeneous twisted differential operators

Let $X,Y$ be smooth complex varieties and let $G$ be an smooth affine algebraic group acting on $X$ and $Y$ such that $X,Y$ are $G$-homogeneous spaces (the $G$ action is transitive). We also let $f:Y \...
4
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2answers
362 views

When does glueing affine schemes produce affine/separated schemes?

Let $X$ be an affine scheme with an open affine subscheme $U\subset X$. Given an automorphism of $U$, we can glue $X$ with itself along $U$ to get a new scheme. Is there a description in terms of ...
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135 views

Proper curve over any base is projective?

Is it true that any proper morphism of relative dimension$\leq 1$ is projective (no additional assumptions whatsoever)? Is it true that any such morphism is $H$-projective (https://stacks.math....
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62 views

Monodromy operators on hyperkähler varieties

Let $X$ be a hyperkähler variety. In an article (Conjecture 2.1) from some years ago, Markman conjectured that any monodromy operator acting trivially on $H^2(X,\mathbb Z)$ is the identity operator, ...
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115 views

Morphisms whose reduction is projective

IIUC Remark 5.3.5 in EGA II says that there exist proper non-projective morphisms $X\rightarrow Y$ where $Y$ is the spectrum of a finite-dimensional $\mathbb{C}$-algebra such that the induced morphism ...
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73 views

On semicontinuity of Hilbert function for a zero dimensional scheme

Let, $X$ be a zero dimensional subscheme of $\Bbb P^n$ and let us define a function as follows: $h_{\Bbb P^n}(X ,d) = \binom {n+d}{n}$ - dim $I_{X}(d)$ , where dim $I_{X}(d)$ = the dimension of ...
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1answer
183 views

$(-2)$-curves in complex $3$-folds

Let $X$ be a smooth complex $3$-fold, and let $C \subset X$ be an embedded smooth rational curve whose normal bundle $N_{C/X}$ is isomorphic to $\mathscr{O}(-1) \oplus \mathscr{O}(-1)$. Is it true ...
4
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0answers
141 views

Fibered surfaces degenerating to Frobenius

Let $R$ be a DVR with algebraically closed, positive characteristic residue field $k$. Let $X\rightarrow Spec(R)$ and $C\rightarrow Spec(R)$ be smooth projective morphisms of relative dimension 2 and ...
2
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1answer
184 views

Purity and skyscraper sheaves

In "The Geometry of moduli spaces of sheaves" a coherent sheaf $\mathcal{F}$ is defined to be pure of dimension $d$ if dim$(\mathcal{E})=d$ for all non-trivial proper subsheaves $\mathcal{E} \subset \...
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0answers
139 views

Kuga-Satake in characteristic $p$ [closed]

Have Kuga-Satake correspondences been investigated in characteristic $p$? (I'm being intentionally vague about what this would mean.)
2
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0answers
82 views

Automorphisms of a neighborhood of a negative curve

Let $X$ e a smooth complex surface and let $C\subset X$ be a smooth rational curve with negative self intersection. Is there any known description of the automorphisms of a infinitesimal ...
5
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0answers
94 views

Open subfunctor of Quot Functor induced by open immersion

Let $f: X \rightarrow S$ be a morphism of noetherian schemes and $\mathfrak{Q}uot_{\mathcal{E}/X/S}$ be the functor parametrizing families of quotients of $\mathcal{E}$ in the category of locally ...
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2answers
206 views

Equivalence between categories of affine schemes over $X$ and representable functors $\operatorname{Points}(x) \to \operatorname{Sets}$

I am reading Strickland - Formal schemes and formal groups. For a functor $X\colon \operatorname{Rings}\to \operatorname{Sets}$, he defines (2.14) the category of $\operatorname{Points}(X)$ in the ...
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0answers
107 views

Topological invariants of a certain “stratified” manifold, with pieces of different “dimensions”

Disclaimer: I don't fully understand what I'm talking about in the question below. I'm still trying to figure out the right question to ask. Quotations and question marks in brackets mean that I'm not ...
5
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0answers
228 views

Obstructions to locally trivial deformations

Let $X$ be a complex projective variety. If $X$ is smooth, then the first-order infinitesimal deformations are given by $H^1(X, T_X)$ and the obstructions are in $H^2(X, T_X)$. Now assume that $X$ is ...
2
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1answer
75 views

Natural morphism to the scheme of isomorphism

Suppose that we have a faithful representation $\rm{G}\rightarrow\rm{GL}(V)$ of a semisimple linear algebraic group into a complex vector space $\rm{V}$ of dimension n. Suppose that we have a ...
1
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1answer
168 views

Is the pushforward of a closed immersion of spectral Deligne-Mumford stacks conservative?

Let $ X \hookrightarrow Y$ be a closed immersion of (connective) spectral Deligne-Mumford stacks, is $ i_* : Qcoh(X) \rightarrow Qcoh(Y)$ conservative? Somehow I couldn't find the statement in SAG...
3
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1answer
138 views

Deformation of stable curve with regular total space [duplicate]

Let $k$ be a field, let $X/k$ be a stable curve. Is it always possible to find a deformation $\mathcal{X}/k[[t]]$ such that $\mathcal{X}$ is regular? (Sorry for the confusion, this is a duplication ...
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0answers
144 views

Proof of Kashiwara's constructibility theorem for algebraic D-modules

I am trying to understand the proof of Kashiwara's constructibility theorem for algebraic D-modules, following either the book "D-modules, Perverse sheaves, and representation theory" by Hotta, ...
1
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0answers
99 views

Is the Bruhat cell Zariski open in a connected algebraic group $G$? [closed]

Is the Bruhat cell Zariski-open in a connected algebraic group $G$? Specifically, is the big Bruhat cell Zariski-open (and maybe Zariski-dense)? Is it true for all the Bruhat cells?
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0answers
124 views

Extend any morphism to suitable projective variety? [closed]

Let $F: X\to \mathbb{P}^n$ be a morphism from an affine variety to projective space (over some algebraically closed field of characteristic zero). Can we always find an open immersion $\iota: X\...
5
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2answers
370 views

About different cohomology theories used to study Shimura varieties

The classical theory of Eichler-Shimura realizes the space of cusp forms of certain weights and levels in the parabolic cohomology of modular curves, which is the image of the cohomology with compact ...
10
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1answer
297 views

When is the non-negative derived category compactly generated?

This question is strongly related to this question. However it seems to me sufficiently distinct to warrant asking it separately. Let $X$ be a quasi-compact, quasi-separated scheme. When is the ∞-...
2
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0answers
96 views

Computing Chow group of a variety which is almost a blow-up of another variety

Let $X$ be a normal complex projective variety (not necessarily smooth), and let $Y$ be a smooth complex projective variety. Let $Z\subset X$ be a smooth closed subvariety. I have a morphism which is ...
2
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0answers
101 views

How to obtain the following “trivial” bound on the number of rational points on a hypersurface?

Let $F: \mathbb{R}^{n} \to \mathbb{R}$ be a smooth function. Suppose $B$ be a closed and bounded box. I would like to obtain for fixed $q \in \mathbb{N}$ $$ \# \{ \mathbf{a} \in \mathbb{Z}^n : F(\...
4
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1answer
671 views

Generalized Behrend version for Grothendieck-Lefschetz trace formula

[MOVED HERE FROM MSE.] The statement of the Grothendieck-Lefschetz fixed point theorem is well-known. For a proper algebraic variety $X$ over $\mathbb F_q$, $$\#X(\mathbb F_q) =\sum_i (−1)^i Tr(Fr_X, ...
7
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0answers
188 views

Adequate equivalence relations and algebraic $K$-theory

I have a somewhat vague question. We know that Adams operation gives a filtration on $K_i(X)\otimes \mathbb{Q}$ for the scheme $X$ such that the weight $j$ elements are isomorphic to higher Bloch Chow ...
1
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1answer
129 views

Log Calabi-Yau surfaces without maximal boundaries

Let $X$ be a smooth projective surface over $\mathbb{C}$, $D\subset X$ is an effective divisor. $(X,D)$ is a log Calabi-Yau pair if $K_X+D$ is a principal divisor. The complement $M=X\setminus D$ is a ...
1
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1answer
123 views

A property for primitive idempotents

Let $R$ be a (commutative) ring (with identity). A nonzero idempotent $e\in R$ is called primitive idempotent, whenever it has no decomposition into $a+b$ where $a$ and $b$ are nonzero orthogonal ($...
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0answers
136 views

Image of quasiprojective variety under closed map

Let $f: X\to Y$ be a regular map of projective varieties that is closed (in the sense that it takes Zariski closed sets to Zariski closed sets). Let $V\subset X$ be a quasiprojective subvariety (i.e. ...
4
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0answers
112 views

Flat base change in the complex analytic setting

On page 255 of Hartshorne's Algebraic Geometry, it is shown that "cohomology commutes with flat base extension": Proposition III.9.3: Let $f : X \to Y$ be a separated morphism of finite type of ...
5
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0answers
167 views

When is vector bundle over toric variety a toric variety?

Is it true that a vector bundle over a toric variety is also a toric variety if and only if it splits? if so, how do we prove it? This seems to be the content of a remark in Oda's Tata's lectures on ...
3
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1answer
160 views

Can a birational map be completed to a proper map?

Let $f\colon X\to Y$ be a birational map of complex algebraic varieties. Are there necessarily open immersions of $X$ and $Y$ into varieties $X’$ and $Y’$, resp., which admit a proper morphism $f’\...