# Questions tagged [ag.algebraic-geometry]

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

15,379
questions

**1**

vote

**0**answers

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 ...

**0**

votes

**0**answers

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, ...

**4**

votes

**1**answer

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 ...

**-2**

votes

**0**answers

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{...

**0**

votes

**1**answer

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 ...

**3**

votes

**0**answers

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 $\overline{X}$ ...

**1**

vote

**0**answers

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 ...

**1**

vote

**0**answers

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 $\pi : Y\rightarrow X$...

**2**

votes

**0**answers

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?

**2**

votes

**0**answers

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 ...

**0**

votes

**1**answer

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**

votes

**0**answers

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 ...

**10**

votes

**0**answers

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 ...

**1**

vote

**0**answers

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**

votes

**1**answer

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 ...

**5**

votes

**0**answers

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....

**2**

votes

**0**answers

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{...

**2**

votes

**0**answers

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**

votes

**2**answers

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 ...

**0**

votes

**0**answers

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....

**2**

votes

**0**answers

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, ...

**1**

vote

**0**answers

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 ...

**2**

votes

**0**answers

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 ...

**6**

votes

**1**answer

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**

votes

**0**answers

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**

votes

**1**answer

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 \...

**2**

votes

**0**answers

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**

votes

**0**answers

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**

votes

**0**answers

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 ...

**1**

vote

**2**answers

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 ...

**1**

vote

**0**answers

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**

votes

**0**answers

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**

votes

**1**answer

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**

vote

**1**answer

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**

votes

**1**answer

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 ...

**7**

votes

**0**answers

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**

vote

**0**answers

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?

**1**

vote

**0**answers

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**

votes

**2**answers

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**

votes

**1**answer

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**

votes

**0**answers

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**

votes

**0**answers

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**

votes

**1**answer

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**

votes

**0**answers

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**

vote

**1**answer

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**

vote

**1**answer

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 ($...

**0**

votes

**0**answers

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**

votes

**0**answers

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**

votes

**0**answers

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**

votes

**1**answer

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’\...