Questions tagged [ag.algebraic-geometry]

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

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How do I compute the compact cohomology of a hypersurface?

How do I compute the compact cohomology of a hypersurface? For example, let $f$ be a Newton polynomial of a polytope in $\mathbb{R}^n$ and let $X = (f=0)$ inside $(\mathbb{C}^\*)^n$ (maybe there is ...
Eric Zaslow's user avatar
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13 votes
5 answers
3k views

Deformations of semisimple Lie algebras

In the questions Is "semisimple" a dense condition among Lie algebras? and What is the Zariski closure of the space of semisimple Lie algebras?, something equivalent to the following is ...
ndkrempel's user avatar
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1 vote
3 answers
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open but not affine subscheme?example? [closed]

In the book algebraice geometry by R.Harshorne we always say :an open affine subset U=SpecA of a sheme X. I was always wondering that whether there existed an open but not affine subset. I want to ...
user11085's user avatar
10 votes
2 answers
689 views

CM for radical ideal

Let $R$ the polynomial ring in $n$ variables with complex coefficients and $I$ an ideal of $R$. Is it true that if $R/I$ is CM also $R/J$ is CM (where $J$ is the radical of $I$)? Is there a relations ...
Michele Torielli's user avatar
2 votes
1 answer
574 views

fundamental group of moduli spaces of sheaves

I am considering moduli spaces of sheaves on irreducible holomorphic symplectic manifolds. I haven't seen a general theory to describe the fundamental group of moduli spaces of sheaves yet. Is there ...
Malte Wandel's user avatar
5 votes
0 answers
179 views

Are $n$-vector bundles an $(\infty,n)$-symmetric monoidal category with duals?

In Lurie's On the Classification of Topological Field Theories, one of the main characters are $(\infty,n)$-symmetric monoidal category with duals. A basic example of this should be $n$-vector spaces, ...
domenico fiorenza's user avatar
2 votes
1 answer
748 views

Endomorphisms of the cohomology of a projective variety

Let $X$ be a smooth projective variety over an algebraically closed characteristic $0$ field $\mathbb{K}$, and let $\Omega^\bullet_X$ be the de Rham sheaf of complexes of $\mathcal{O}_X$-modules of ...
domenico fiorenza's user avatar
8 votes
3 answers
3k views

The automorphism group of a hyperelliptic curve

Let $C$ be the projective smooth genus 2 curve defined by $y^2=x^5-x$ over $\mathbb F_5.$ What is the order of its automorphism group (automorphisms over $\mathbb F_5$)? I have seen different ...
shenghao's user avatar
  • 4,195
7 votes
3 answers
981 views

Quantum cohomology rings as invariants

Let $X$ and $Y$ manifolds. What kind of relations between them (like homeomorphism, diffeomorphism, homotopy equivalence) gives an isomorphic quantum cohomology rings?
Cat's user avatar
  • 71
12 votes
4 answers
2k views

extension of $G$-bundles

Let $S$ be a smooth surface (let's say over an algebraically closed field) and let $D$ be a smooth divisor in $S$. Let also $G$ be a connected algebraic group. Assume that we are given a principal $G$-...
Alexander Braverman's user avatar
1 vote
1 answer
355 views

Cancellation problem for curves

I recently stumbled upon a paper (NON CANCELLATION FOR SMOOTH CONTRACTIBLE AFFINE THREEFOLDS) about the cancellation problem: If $X$ is a variety over $\mathbb{C}$ of dimension $d$ such that $X \times ...
solbap's user avatar
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4 votes
2 answers
517 views

Non-Ruled Minimal Surfaces

Comrades, Let $S$ be a non-ruled, minimal (smooth projective complex algebraic) surface. Let $K$ be a canonical divisor of $S$ and $H$ a hyperplane section of $S$ (for your favorite embedding). ...
Robert Garbary's user avatar
16 votes
1 answer
2k views

Universal homeomorphisms and the étale topology

Let $f:X\to S$ be a universal homeomorphism of schemes. Assume $X(S')\neq\emptyset$ for some étale surjective $S'\to S$. Does $f$ have a section? The answer is yes if $S$ is reduced, by descent. ...
Laurent Moret-Bailly's user avatar
9 votes
2 answers
796 views

Is projectivity local on the base?

Let $f:X\to Y$ be a morphism of schemes and assume that $Y$ has an open cover $\{U_i\}$ such that $f:f^{-1}U_i\to U_i$ is projective. Does it follow that $f$ is projective?
Alfonz's user avatar
  • 191
5 votes
1 answer
733 views

trivial subbundles

Let X be a projective variety (say, irreducible) and E a vector bundle on X or rank r. Is it true that there exists a codimension 2 closed subset Z in X such that restriction of E(n) (for n large ...
Vladimir Baranovsky's user avatar
1 vote
1 answer
563 views

Sheaf of differential and its reflexive hull on a toric variety

Let $X$ be a non-smooth toric variety, $\Omega_X$ be the sheaf of differentials, $\hat{\Omega}_X$ the double dual of $\Omega_X$. My questions are: Is there any chance that $\Omega_X=\hat{\Omega}_X$? ...
Zhiyu's user avatar
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5 votes
1 answer
313 views

When is the projective line the seminaive projective line?

Excuse the possible naivete of this question. Since reading a nice survey article by Daniel Biss a few years ago, I'm always worried about what $P^1(R)$ is, for a ring $R$. So that I stop worrying, ...
Marty's user avatar
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7 votes
1 answer
751 views

Schemes (as in algebraic geometry) and first-order logic.

Affine schemes are simply the Zariski spectra of commutative rings, and commutative rings occurs as models of a first-order theory. I would guess that general schemes do not naturally correspond to ...
David Feldman's user avatar
5 votes
2 answers
2k views

Exterior derivative on almost complex manifolds

Let $M$ be a complex manifold, and $\omega$ be a $(p,q)$-form. Then $d\omega$ is an element of $\Omega^{p+1,q}(M)\oplus\Omega^{p,q+1}(M)$, so that $d = \partial + \overline{\partial}$, where $\partial$...
miramo's user avatar
  • 515
11 votes
2 answers
902 views

Homologically nice commutative rings

Let $R$ be a commutative regular local ring. Is it true that for every $\mathfrak p \in \mathrm{Spec}(R)$ there is a finitely generated $R$-module $M$ such that $\mathrm{projdim}(M)=\mathrm{ht}(\...
David's user avatar
  • 203
3 votes
0 answers
186 views

affineness vs geometric affineness

Let $X$ be $k$-scheme of finite type, $k$ being a (perfect) field, and assume $X\otimes\overline{k}$ is affine. Is $X$ necessarily an affine scheme? What about if $X$ is a $k$-group scheme?
shenghao's user avatar
  • 4,195
13 votes
1 answer
560 views

What is the second fundamental form of moduli space?

Away from the hyperelliptic locus, the moduli of curves immerses in the moduli of principally polarized abelian varieties. The ambient space has a riemannian metric, so one can ask about the second ...
Ben Wieland's user avatar
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-1 votes
1 answer
523 views

Relative flatness

Can someone me say if this (perhaps obvious!) claim is true: let $f:X\rightarrow S$ be an open, surjective morphism of complex spaces reduced or without embedded components and with $n$-pure ...
kaddar's user avatar
  • 435
2 votes
1 answer
582 views

Computations in the Deligne-Mumford compactification with marked points

The following question concerns Deligne-Mumford compactification of the (coarse) moduli space $M_{g,n}$ of smooth complex genus g curves with $n$ marked points. If there are no marked points (ie $n=...
Alex's user avatar
  • 454
0 votes
2 answers
316 views

Quasiprojectiveness of bundle

Let $X$ be a quasiprojective variety(all varieties are over a field $k$). Is an algebraic vector bundle $E$ over $X$ quasiprojective? If $X$ is affine, is $E$ affine? Is a projective ...
liu hang's user avatar
5 votes
1 answer
2k views

Intersections of irreducible components

Let $V$ be an algebraic variety (not irreducible) over $\mathbb{C}$, defined by an ideal $I = \{f_1,f_2,\dots, f_n\}$. $V$ is not necessarily pure dimensional. Suppose $V = R_1\cup R_2\cup\dots\cup ...
Brian's user avatar
  • 1,502
1 vote
1 answer
736 views

Application of the base change theorem

Given a flat and projective morphism $f:X\rightarrow Y$ of noetherian schemes over some algebraically closed $k$ and $F$, $G$ coherent $O_X$-modules, flat over $Y$. Then the base change theroem for ...
TonyS's user avatar
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1 vote
1 answer
642 views

A composition of a finite morphisms with the transpose correspondence: is it the multiplication by the degree?

Let $X,Y$ be smooth varieties over a field $k$ (which in my case is perfect of finite characteristic $p$; we may also assume that $X,Y$ are connected); $s:Y\to X$ is a finite morphism of degree $d$. ...
Mikhail Bondarko's user avatar
7 votes
3 answers
1k views

How many independent quadrics should one intersect to get the canonical curve.

Let $C$ be a non hyperelliptic complex algebraic curve of genus $g$, then the vector space $I_2(C)$ of quadrics containing the canonical image of $C$ is $\binom{g+1}{2}-h^0(2\omega_C) = (g-2)(g-3)/2$ ...
David Lehavi's user avatar
  • 4,299
3 votes
2 answers
706 views

Does monodromy act on the derived category of sheaves?

Let $\mathcal{X} \to \Delta^* $ be a family of complex projective varieties over a punctured disc. Then, for any fibre $X$, there is a monodromy action $M: H^* (X) \to H^*(X)$. Is there a ...
Vivek Shende's user avatar
  • 8,663
14 votes
1 answer
2k views

Is the cotangent bundle to a Kahler manifold hyperkahler?

Let me be more specific. Let $M$ be a Kahler manifold with Riemannian metric $g$ and complex structure $I$. Then $T^\ast M$ will also be Kahler with metric and complex structure induced from $M$ (I ...
Sam Gunningham's user avatar
30 votes
1 answer
4k views

Torsors for finite group schemes

Let $k$ be a field of characteristic $p > 0$ (assume $k$ is perfect if it helps). Let $G$ be a connected finite group scheme of height one over $k$. Then $G$ is determined by its Lie algebra $\...
Jacob Lurie's user avatar
  • 17.5k
3 votes
1 answer
821 views

A form of cohomology and base change

Let $f \colon X \to Y$ be a proper morphism of (Noetherian) schemes, $\mathcal{F} \in \mathop{Coh}(X)$. Let $i_Z \colon Z \hookrightarrow Y$ be a closed subscheme and take the inverse image $W := X \...
Andrea Ferretti's user avatar
8 votes
3 answers
2k views

When is a blow-up along a union of subvarieties smooth?

Let $V_1$ and $V_2$ be two distinct smooth subvarieties of the smooth variety $X$ which are regularly embedded. I would like to find a reasonable criteria which guaranties the smoothness of the Blow-...
Passenger's user avatar
  • 680
1 vote
1 answer
2k views

Quartic curve - what is the genus?

I am studying the following quartic curve: $f(x,y) = c_1x^2 + c_2x^4 + c_3x^2y + c_4x^2y^2 + c_5y^2 + c_ 6y^3 + c_7y^4$ where $c_i$ are constant (in fact they are expressions in terms of other ...
Rodrigo's user avatar
  • 11
22 votes
1 answer
1k views

What is classified by the (big) crystalline topos?

In his paper "Generic Galois Theory of Local Rings", G.C. Wraith states on p. 743 that the (big) crystalline topos "can be conveniently described in terms of the theory it describes". What exactly is ...
Marc Nieper-Wißkirchen's user avatar
18 votes
6 answers
2k views

Non-Kahler "Calabi-Yau"?

Are there examples of non-Kahler complex manifolds with holomorphically trivial canonical bundle?
680's user avatar
  • 385
7 votes
1 answer
680 views

Zeta function of monodromy and counting points over C((t))

If $X$ is a smooth, projective variety over $\mathbb{F}_q$, the Weil conjectures tell us: $$\prod \mathrm{det} (I - TF|_{H^i_c(X)})^{(-1)^{i+1}} = \mathrm{exp}\left(\sum_{m=1}^{\infty} \frac{N_m}{m} ...
Vivek Shende's user avatar
  • 8,663
3 votes
1 answer
249 views

closed substacks of cartesian powers of a stack

Let $\mathbb{Z}/2\mathbb{Z}$ act on $\mathbb{A}^1$ as $x \mapsto -x$, and let $\mathscr{X}$ be the quotient stack. It has coarse moduli space $\mathbb{A}^1$ and a residual gerbe $B\mathbb{Z}/2\mathbb{...
Dima Sustretov's user avatar
16 votes
0 answers
1k views

Can one compare integral structures on de Rham and crystalline cohomology?

Suppose $\mathfrak{X}$ is a smooth projective scheme of finite type over $\mathbb{Z}_p$, with generic fibre $X$. Then there are comparison theorems relating de Rham and crystalline cohomology, $H^i_{\...
David Loeffler's user avatar
8 votes
3 answers
1k views

relation between toric geometry and log geometry

Hello, I'm trying to understand the relation between the points of view of log geometry (monoids) and toric geometry (fans). Suppose that $k$ is a field and $P$ is a finitely generated monoid. Then $...
unknown's user avatar
  • 647
0 votes
1 answer
685 views

Global Spec and Vector Bundles

Can anybody explain why the vector bundle corresponding to a locally free sheaf F is global spec of sym of the dual of F and not just F? How does a section get identified with a polynomial in the ...
Victor's user avatar
  • 19
0 votes
1 answer
257 views

Subtleties in the construction of base change morphisms

Given a flat and projective morphism of noetherian schemes, $f: X \rightarrow Y$ and $F$, $G$ two coherent $O_X$-modules, flat over $Y$. Furthermore given a morphism $u: Y' \rightarrow Y$ of ...
TonyS's user avatar
  • 1,391
0 votes
1 answer
284 views

Chow group of a fiber product of grassmann bundles

Let $X$ be a smooth projective variety and $E\longrightarrow X$ a vector bundle of rank $n$. For any $0\leq k\leq n$ the associated Grassmann bundle $G_k(E)\longrightarrow X$ yields and we have the so-...
Charles's user avatar
  • 193
4 votes
0 answers
257 views

semisimple orbifold quantum cohomology

Dubrovin conjecture says, roughly speaking, that the quantum cohomology of a variety $X$ is semisimple if and only if it is a good Fano [good means that there exits a full exceptional collection in ...
Cat's user avatar
  • 41
6 votes
1 answer
705 views

Hyperbolicity for algebraic varieties and relation to curves on them

My question is related to several notions of hyperbolicity, applied to Kähler manifolds (projective, in general). Kähler hyperbolicity was introduced in this paper of Gromov's. He calls a Kähler ...
sfilip's user avatar
  • 273
11 votes
2 answers
2k views

Non-Kahler Complex manifolds

For a non-Kahler complex manifold $M$, we still have the decomposition of differential forms into differential forms of type $(p,q)$ and we can write $d=\partial+\bar\partial$ and we can define ...
Mohammad Farajzadeh-Tehrani's user avatar
14 votes
6 answers
3k views

Generalizations of Belyi's theorem

Belyi's theorem states that the following properties of a nonsingular projective algebraic curve $X$ are equivalent: 1) $X$ is defined over $\overline{\mathbb{Q}};$ 2) There exists a meromorphic ...
1 vote
1 answer
696 views

Solving nonlinear ODE's

If I would ask for $\phi'(x) = f( \phi(x))$ and $\phi(0)=f(0)$, I would get that the inverse of $\phi$ is forced to be of the form: $$\phi^{-1}(z) = \int_{0}^z \frac{1}{f(x)} d x.$$ Now it is ...
2 votes
1 answer
543 views

Numerically rigid nef divisor

Is it possible to find an example of an $\mathbb{R}$-Cartier divisor $D$ on an irreducible variety $X$ that is non-trivial, nef, effective and numerically rigid? By "numerically rigid" I mean that ...
Oren's user avatar
  • 195

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