Questions tagged [ag.algebraic-geometry]
Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology.
21,545
questions
11
votes
1
answer
1k
views
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 ...
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 ...
1
vote
3
answers
2k
views
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 ...
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 ...
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 ...
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, ...
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 ...
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 ...
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?
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 $G$-...
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 ...
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).
...
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. ...
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?
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 ...
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 $\Omega_X=\hat{\Omega}_X$?
...
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, ...
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 ...
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 $\partial$...
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}(\...
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?
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 ...
-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 ...
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=...
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 ...
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 ...
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 ...
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 $d$. ...
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 $\binom{g+1}{2}-h^0(2\omega_C) = (g-2)(g-3)/2$ ...
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 ...
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 ...
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 $\...
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 \...
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-...
1
vote
1
answer
2k
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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
...
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 ...
18
votes
6
answers
2k
views
Non-Kahler "Calabi-Yau"?
Are there examples of non-Kahler complex manifolds with holomorphically trivial canonical bundle?
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} ...
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{...
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_{\...
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 $...
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 ...
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 ...
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-...
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 ...
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 ...
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 ...
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 ...