Questions tagged [ag.algebraic-geometry]
Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology.
21,545
questions
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 ...
5
votes
2
answers
351
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 $Y$ ...
7
votes
1
answer
745
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 ...
8
votes
1
answer
595
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 ...
8
votes
1
answer
287
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
votes
1
answer
383
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 ...
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})...
0
votes
1
answer
287
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?
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 $...
1
vote
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 ...
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 ...
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)?
3
votes
0
answers
140
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 ...
2
votes
1
answer
367
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, ...
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 ...
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 ...
4
votes
1
answer
305
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}...
5
votes
1
answer
715
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" ...
4
votes
1
answer
886
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 $Y$ ...
4
votes
0
answers
150
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". ...
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=...
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 $\mathcal{O}_X$-...
8
votes
2
answers
868
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 ...
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 ...
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 $[\...
6
votes
0
answers
160
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 \...
11
votes
1
answer
695
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 ...
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 ...
2
votes
0
answers
149
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 ...
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 $X$?
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 $\mathscr{X} : \mathsf{CAlg}(k) \to \mathsf{Gpd}$ (...
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 ...
3
votes
1
answer
577
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 ...
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}...
10
votes
0
answers
684
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 ...
2
votes
1
answer
422
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 ...
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 ...
2
votes
2
answers
840
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 $X$ ...
7
votes
1
answer
423
views
Understanding a germ of a GIT quotient
Let $X$ be a smooth complex affine variety, let $G$ be a complex reductive group acting on $X$. Suppose that the stabilizer $G_x$ of a point $x\in X$ is reductive and connected. Let $\varphi: X\to X//...
44
votes
2
answers
3k
views
Bijection from the plane to itself that sends circles to squares
Let me apologize in advance as this is possibly an extremely stupid question: can one prove or disprove the existence of a bijection from the plane to itself, such that the image of any circle ...
4
votes
1
answer
323
views
A sub-variety of a Grassmannian
Is there a nice description of the variety $G(r,2r) \setminus \sqcup_{i+j=r}(G(i,r) \times G(j,r))$ in terms of blow ups or a sub-variety of a secant variety or any other natural construction to see ...
5
votes
1
answer
334
views
Is there a relationship between the moduli space of spatial polygons and the moduli space of labeled points?
It is well known that the set of all polygons with consecutive side lengths $l_1, \dots, l_n$ in $\mathbb{R}^3$, considered up to rigid motions, is a compact complex manifold. Of course, I am assuming,...
2
votes
0
answers
249
views
Proj construction and pushforward of line bundles
Let $X$ be a variety of dimension $d \geq 2$ (over a field), consisting of two irreducible components meeting transversely in a divisor $D$. (We can assume these components and $D$ are as nice as we ...
8
votes
1
answer
803
views
There are no "holes" in the Bruhat decomposition of parabolic cell $Pw_1P$
Let $G$ be a split reductive algebraic group (over a local field if you like), $B$ be a fixed Borel subgroup, and $P$ be a fixed standard parabolic subgroup. Let $W$ be the Weyl group of $G$. For $w\...
0
votes
1
answer
248
views
Analytic spread of localization of an ideal
Let $J$ be an ideal in a Noetherian local ring $(R,m)$. It is well known that for any prime ideal $p\in Spec(R)$, $l(J_p)\leq l(J)$, where $l(J)$ is the analytic spread of $J$.
Q) Are there ...
2
votes
1
answer
364
views
Involutions on $\mathbb{C}(x,y)$
How to find all involutions on $\mathbb{C}(x,y)$,
or at least all involutions $\delta$ on $\mathbb{C}(x,y)$ such that
$\delta(x)=x$?
Remarks:
(1) An involution on $\mathbb{C}[x,y]$ is either ...
5
votes
0
answers
281
views
On rational points in an affine space
Let $X$ be a quasi-projective scheme over $\mathbb{F}_q$. If its base change to $\overline{\mathbb{F}}_q$ is an affine space $\mathbb{A}^n$, then it is known that $|X(\mathbb{F}_q)|=q^n$.
The only ...
5
votes
0
answers
123
views
Degree of unirational parametrization of hypersurfaces
Let $X_d\subset \mathbb P^{n+1}_{\mathbb C}$ be a smooth hypersurface of ddegree $d$. Harris, Mazur and Pandharipande proved that there is a bound $b(d)$ such that if $n>b(d)$, $X_d$ is unirational....
3
votes
0
answers
160
views
An explicit description of $A_X$
Let $X$ be a smooth complex projective curve of genus at least $2$ and $p\in X$. Define $$A_X=H^0(X-p,\mathcal O_X).$$
By choosing a local parameter $z$ near $p$ we can see $A_X\subset Frac(\hat{O}_{X,...
5
votes
0
answers
293
views
Resolving analytic normal crossings singularities
Let $X$ be a non-singular (complex) variety and $Y \subset X$ be a (reduced) irreducible subvariety with only normal crossings singularity (locally, in the analytic topology, the singularity is ...