Questions tagged [ag.algebraic-geometry]
Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology.
21,607
questions
3
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0
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364
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Fujita decomposition versus Zariski decomposition
Fujita decomposition: Let $\frak \pi : X \to B$ be a fibration of a compact Kahler manifold $\frak X$ over a projective curve $\frak B$ then $\pi_*\left(K_{\frak X/B}\right)=A\oplus B$ where $A$ is ...
user21574
4
votes
1
answer
647
views
Infinitely many exceptional curves on ruled surfaces
Take an algebraically closed field $k$. Let $C$ be a smooth projective variety of dimension $1$ over $k$(a curve). Consider a geometrically ruled surface $S :=C\times\mathbb{P}^1$. Does there exists a ...
- 205
1
vote
0
answers
96
views
Intersection of two semiorthogonal decompositions on a variety
Let $X$ be a smooth, projective variety, and $E_1$ and $E_2$ be two smooth divisors on it meeting each other normally. Suppose that $D^b_{E_1}(X) \subset D^b(X)$ and $D^b_{E_2}(X) \subset D^b(X)$ are ...
- 291
3
votes
1
answer
194
views
Why are the toric fibers of a toric manifold Lagrangian submanifolds?
How does one know in general that the toric fibers of a Kahler toric manifold are Lagrangian submanifolds? I can roughly fathom this for e.g. a circle in $\mathbb{C}P^1$, but how about more general ...
- 1,135
3
votes
0
answers
320
views
Ring of invariants and Borel subgroup
Let $G$ be a connected algebraic group (can assume $G$ to be reductive) acting on a $k$-algebra $A$. Let $B$ be a Borel subgroup of $G$.
Q. Is it generally true that the the ring of invariants $A^...
- 113
15
votes
2
answers
2k
views
Meaning of the determinant of cohomology
The Arakelov intersection number on arithmetic surfaces is defined as an "extension" of the classical intersection number on algebraic surfaces. It was introduced to get a nice intersection theory ...
- 299
7
votes
0
answers
658
views
High dimensional analogue of Ramanujan's pi formula
The question below comes to my mind when I am trying to explore something related to the formulas found by Jesus Guillera:
a)Generalized hypergeometric function
$${}_3 F_2\left(\begin{matrix}1/4&...
- 3,317
4
votes
1
answer
2k
views
Is every projection birational morphism between integral schemes a blowing up morphism?
I apologize if this question is too basic, but I figure this should be an easy question to answer for experts.
Theorem 8.1.24 in Qing Liu's "Algebraic Geometry and Arithmetic Curves" says:
"Theorem ...
- 7,161
2
votes
0
answers
98
views
Definition of an attractor of a stack under an action of $G_{m}$
For an algebraic space $Z$ with the action of multiplicative group scheme $G_{m}$ one can define the attractor space $Z^{+}$ as the functor which sends a scheme $S$ to the set $Map(S \times A^{1},Z)^{...
- 21
2
votes
1
answer
197
views
On the notion of sub-hyperfield
I am working on some generalization of the paper of Gel'fand, Rybnikov and Stone "Projective orientations of matroids" to the wide context of matroids over hyperfields.
I would like to now if the ...
- 741
1
vote
1
answer
244
views
The geometric genus under a generically finite to one rational map [closed]
Is it true that if $X$ and $Y$ are two irreducible algebraic curves over $\mathbb{C}$ and $f:X\dashrightarrow Y$ is a rational map that is generically n:1 for some $n\in \mathbb{Z}^{+}$ then the ...
- 13
8
votes
1
answer
883
views
Algebraic fundamental group of a variety
I have a very explicit question. Consider a projective variety (a Fano 3-fold) in $\mathbb P^{10}$ defined by 3 quadrics and 32 cubic equations. I want to show that the algebraic fundamental group of ...
- 81
2
votes
0
answers
68
views
Centralizer/Normalizer of global sections of vector bundles on curves
Let $X$ be a smooth, projective curve of genus at least $2$ over $\mathbb{C}$ and $E$ be a vector bundle on $X$ of rank at least $2$. Given any point $x \in X$, denote by $S_x$ the image of the ...
- 1,949
5
votes
2
answers
352
views
fibers of birational contraction for complex manifolds - are they Moishezon?
Let $X$ be a smooth complex manifold and
$\phi:\; X \mapsto Y$ a proper holomorphic
map which is birational ("birational contraction"),
and $Z= \phi^{-1}(y)$ its fiber in a point $y$.
The variety $Y$ ...
- 9,095
7
votes
1
answer
747
views
Special cases of the Kimura-O’Sullivan conjecture, i.e., examples of finite dimensional motives?
In this 2005 paper, Kimura introduces a notion of finite dimensionality for Chow motives, defined in terms of vanishing of high symmetric and wedge powers. Toward the end of his paper, he conjectures ...
- 944
8
votes
1
answer
598
views
Number of connected components of an Automorphism group
Let $X$ be a smooth quasi-projective irreducible variety over the field of complex numbers $\mathbb{C}$. We denote by $\mathrm{Aut}(X)$ the group of algebraic automorphisms of $X$. Moreover, for a ...
- 413
8
votes
1
answer
289
views
Families of curves on compact complex surfaces and algebraicity
Let $S$ be a compact complex manifold of dimension $2$ and assume that there exists a two-dimensional family of curves on $S$. Is it true then that the algebraic dimension of $S$ is $2$, i.e. that $S$ ...
- 4,060
4
votes
1
answer
385
views
Endomorphism of globally generated sheaves on curves
Let $X$ be a smooth, projective curve (over $\mathbb{C}$) of genus at least $2$ and $E$ be a globally generated sheaf on $X$. I am looking for conditions/examples such that there exists a closed point ...
- 1,949
2
votes
0
answers
142
views
Bidouble covers and the pushforward of the canonical bundle
In his paper "Singular bidouble covers and the construction of interesting algebraic surfaces", Catanese gives the following theorem
Theorem: Let $f:Y\to X$ be a finite flat $(\mathbb{Z}/2\mathbb{Z})...
- 363
0
votes
1
answer
289
views
Left and right $t$-structures
Several sources I see speak of a "left t-structure", but lack a precise definition. Where can I find a reference for this?
- 13.2k
4
votes
1
answer
215
views
Do only finitely many bisecants of a canonical curve intersect two distinct codimension 2 spaces simultaneously?
Setup & question
Let $C \hookrightarrow \mathbb{P}^{g-1}$ be a general canonical curve of genus $g \ge 4$ and let $Y_1,Y_2 \subset \mathbb{P}^{g-1}$ be codimension 2 linear subspaces such that $...
- 813
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0
answers
182
views
Can we choose $h(\mathcal{O})$ elliptic curves such that they have same trace and endomorphism ring $\mathcal{O}$ but distinct j-invariants?
Let $\mathbb{F}_q$ be a finite field of characteristic $p$ with $q=p^a$ elements. Let $E$ be an ordinary elliptic curve defined over $\mathbb{F}_q$. We know that $\text{End}(E)\otimes\mathbb{Q}$ is ...
- 139
3
votes
0
answers
80
views
Singularities of fibrations 2
This question is related to my previous question:
Singularities of fibrations
Assume that $X$ is a complete intersection irreducible $3$-fold in a product of projective spaces. So that $X$ is ...
- 8,852
6
votes
1
answer
243
views
Is $\mathbb{CP}^2$ with a line collapsed a complex analytic space?
Consider the quotient space of $\mathbf{CP}^2$ obtained by collapsing a line (a $\mathbf{CP}^1$) to a point. Is this a complex analytic space (in a natural way)?
- 257
3
votes
0
answers
141
views
localization of ringed spaces
A primed ringed space is a ringed space $X$ equipped with a prime system $M$, which by definition is a map assigns to each point $x\in X$ a subset $M_x\subseteq {\rm Spec}\, \mathcal{O}_{X,x}$.
In ...
- 554
2
votes
1
answer
370
views
Homotopy groups of noncommutative spaces
In the approach to noncommutative geometry of Alain Connes any Hausdorff compact space $X$ is replaced by its algebra of complex valued continuous functions $C^0(X)$, and one regard general (that is, ...
user95283
4
votes
1
answer
2k
views
Irreducibility of fiber product of irreducible varieties via dominant morphisms
Let $X,Y,Z$ are irrreducible varieties. $f:X\to Y$ is prpoer surjective and $g:Z \to Y$ is dominant.
Then, $X\times_Y Z$ is irreducible?
Moreover, it will be very helpful for me if there are other ...
- 421
5
votes
0
answers
145
views
Injectivity of a standard map in quiver representation
Let $X$ be a smooth projective variety, and assume its divisor class group is finite and free. Let $E_1,E_2,\ldots,E_n$ be line bundles on $X$. Define $L_k=E_1+\ldots E_k$, and let $Q$ be the ...
- 815
4
votes
1
answer
306
views
Singularities of fibrations
Let $f:X\rightarrow \mathbb{P}^2$ be a fibration, here $X$ is a projective variety of dimension three.
Assume that there exixts a smooth curve $C\subset\mathbb{P}^2$ such that for any $p\in\mathbb{P}...
- 8,852
5
votes
1
answer
717
views
To derive or not to derive, that is the question
What are concrete and abstract examples of problems (even whole programmes of inquiry) when one has a choice to use a "derived" theory (e.g., $\infty$-categories, DAG, HAG, $DRep_k(G)$, "higher" ...
- 629
4
votes
1
answer
890
views
Regular functions on a product of varieties
Let $k$ be an algebraically closed field and let $X$, $Y$ be varieties over $k$.
Let us denote by $\mathcal{O}(X)$ and $\mathcal{O}(Y)$ the $k$-algebra of regular functions on $X$ and $Y$ ...
- 413
4
votes
0
answers
151
views
Trivialization of a fibration after a base change
Let $E\to B$ be a locally trivial fibration of curves of genus >1 (everyhting is compact and over $\mathbb{C}$). I read that this fibration becomes trivial "after a finite unramified base change". ...
- 41
2
votes
1
answer
162
views
Singularities of $3$-folds
Let $X,Y,Z$ be projective $3$-folds. Assume that $Y$ is smooth and $Z$ is smooth and Fano. Moreover, assume that there is a generically finite morphism $f:Y\rightarrow Z$ admitting a factorization $f=...
user97096
7
votes
0
answers
356
views
Free modules generate all quasi-coherent modules
The following statement is true* and not hard to prove.
Let $X$ be a quasi-compact and separated scheme. Then every quasi-coherent $\mathcal{O}_X$-module is a subquotient of a free $\mathcal{O}_X$-...
- 61.5k
8
votes
2
answers
873
views
Unipotent algebraic group action on quasi-affine (vs. affine) variety?
This question arises from a comment by user nfdc23 on an unrelated recent MO question here. It concerns textbook treatments of what has been called the "Theorem of Kostant-Rosenlicht", stated as ...
- 52.4k
8
votes
1
answer
569
views
Smoothing of a Kähler orbifold metric on a complex surface
Let $S$ be a smooth complex projective surface and $D\subset S$ be a smooth complex curve. Fix an integer $m>1$ and consider $(S,D,m)$ as an orbifold with orbi-locus $D$ with stabilizer $\mathbb ...
- 14k
11
votes
3
answers
921
views
What does it mean that $[X]+[Y]=0$ in the Grothendieck ring of varieties?
This is a really basic question. If I have two non-isomorphic varieties $X$ and $Y$, is it possible that $[X]+[Y]=0$ in the Grothendieck ring?
If so, what does this mean geometrically? Obviously $[\...
- 2,768
6
votes
0
answers
161
views
How does the "todd class operator" commute with Nakajima's q operators on Hilbert schemes of points on surfaces
Let $S$ be a surface, and $S^{[n]}$ the Hilbert scheme of $n$ points on $S$. One defines
$$
\mathbb H = \bigoplus_n H^*(S^{[n]}).
$$
One has Nakajima's operators $\mathfrak q_n(\alpha)$ (with $\alpha \...
- 1,469
11
votes
1
answer
697
views
Chow ring of Hilbert scheme of 4 points in $\mathbb{P}^2$
What is known about the Chow ring of the Hilbert scheme of length 4 subschemes of $\mathbb{P}^2$?
I know there is work on cycles on Hilbert schemes in the literature, but I don't know what can be ...
- 1,537
4
votes
0
answers
144
views
tangent vectors defined by smooth curves
I would like to know under which (minimal) conditions an irreducible complex algebraic variety $X$ has the following property: given a closed point $P\in X$, the tangent space $T_PX$ of $X$ at $P$ is ...
- 729
2
votes
0
answers
150
views
toroidal compactifications of modulis spaces of ppav's
Are the modular toroidal compactifications of ppav's (second Voronoi) defined by Alexeev without self-intersections? i.e. are the irreducible component of the boundary divisor normal? If not, can one ...
- 337
14
votes
1
answer
690
views
If $X\times X$ is rational, must $X$ also be rational?
Is there an example of a smooth projective variety $X$ such that $X$ is irrational, but $X\times X$ is rational?
For instance, is $X\times X$ irrational for a smooth cubic threefold $X$?
- 666
4
votes
0
answers
227
views
Quasi-coherent module of (global) finite presentation
If $\mathscr{X}$ is a stack over some base ring $k$ (if you are not familiar with stacks, read "schemes" here), we may consider it as a pseudofunctor $\mathscr{X} : \mathsf{CAlg}(k) \to \mathsf{Gpd}$ (...
- 61.5k
8
votes
1
answer
402
views
Is $H_{et}^1(X,F) = H^1(\pi_1^{et}(X), F(\bar{k}))$ true?
Let $X$ be a smooth geometrically connected scheme over a field $k$ of characteristic 0 (but not necessarily algebraically closed, I can take it to be a number field). Let $F$ be a finite algebraic ...
- 83
3
votes
1
answer
582
views
When every ideal containing $J(R)$ is an intersection of maximal ideals
Let $R$ be a commutative ring with $1$ such that every ideal containing $J(R)$, the intersection of all maximal ideals, is an intersection of maximal ideals. Is there any characterization for such a ...
- 33
8
votes
1
answer
2k
views
Deligne's Canonical Extension in Algebraic Varieties?
Suppose $C/\mathbb{Q}$ is an algebraic curve (not necessarily complete) defined over $\mathbb{Q}$, and $p$ is a $\mathbb{Q}$ valued point of $C$. Suppose there is a algebraic fibration
\begin{equation}...
- 2,961
10
votes
0
answers
688
views
Mumford's intuition for flatness
In Mumford's book Algebraic Geometry II, he writes on page 179..."In order to get at what I consider the intuitive content of "flat" we need first a deeper fact..."
After the deeper fact is proven he ...
- 431
2
votes
1
answer
435
views
Open and Dense Substack
I am looking for a definition of open and dense substack of a Deligne-Mumford stack $\mathcal X$. I have encountered this notion many times, but I am not able to find any references in which dense ...
- 123
2
votes
1
answer
150
views
Analytic germ of a GIT quotient at a fixed point
Let $X$ be a smooth complex affine variety with an action of a complex reductive group $G$. Suppose that $x$ is a fixed point. Denote by $\varphi$ the GIT quotient $\varphi: X\to X//G$.
Question. How ...
- 14k
2
votes
2
answers
844
views
Rational maps and Kodaira dimension
Let $\phi:X\dashrightarrow Y$ be a generically finite, dominant rational map between smooth projective varieties over $\mathbb{C}$.
Assume that $Y$ is of general type. May we conclude then that $X$ ...
- 8,852