Questions tagged [ag.algebraic-geometry]

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

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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 ...
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2 answers
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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 $Y$ ...
Misha Verbitsky's user avatar
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 ...
Tyler Foster's user avatar
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 ...
Anonymous's user avatar
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8 votes
1 answer
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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 $S$ ...
Dima Sustretov's user avatar
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 ...
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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})...
pjox's user avatar
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1 answer
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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?
fosco's user avatar
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4 votes
1 answer
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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 $...
Emre's user avatar
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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 ...
tiansong's user avatar
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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 ...
Puzzled's user avatar
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6 votes
1 answer
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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)?
Alex's user avatar
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0 answers
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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 ...
G.-S. Zhou's user avatar
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, ...
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4 votes
1 answer
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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 ...
keaton's user avatar
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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 ...
Xuqiang QIN's user avatar
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}...
Puzzled's user avatar
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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" ...
Artur Jackson's user avatar
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 $Y$ ...
Anonymous's user avatar
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0 answers
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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". ...
Andrew's user avatar
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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=...
user avatar
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 $\mathcal{O}_X$-...
Martin Brandenburg's user avatar
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 ...
Jim Humphreys's user avatar
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 ...
aglearner's user avatar
  • 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 $[\...
user2520938's user avatar
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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 \...
Drew's user avatar
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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 ...
DCT's user avatar
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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 ...
Hephaistos's user avatar
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 ...
cata's user avatar
  • 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 $X$?
byu's user avatar
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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 $\mathscr{X} : \mathsf{CAlg}(k) \to \mathsf{Gpd}$ (...
Martin Brandenburg's user avatar
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 ...
user108289's user avatar
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 ...
Anderias's user avatar
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}...
Wenzhe's user avatar
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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 ...
Karl's user avatar
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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 ...
Symòn's user avatar
  • 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 ...
aglearner's user avatar
  • 14k
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 $X$ ...
Puzzled's user avatar
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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//...
aglearner's user avatar
  • 14k
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 ...
Tom Solberg's user avatar
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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 ...
icmes's user avatar
  • 43
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,...
Priyavrat Deshpande's user avatar
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 ...
user106980's user avatar
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\...
Q. Zhang's user avatar
  • 960
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 ...
Cusp's user avatar
  • 1,703
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 ...
user237522's user avatar
  • 2,783
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 ...
user148212's user avatar
  • 1,534
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....
pi_1's user avatar
  • 1,433
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,...
7100note4's user avatar
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 ...
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