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

learn more… | top users | synonyms (1)

0
votes
0answers
112 views

Total degree of a polynomial

Let $\mathsf{F,G}\in\Bbb R[x_1,\dots,x_n]$ be minimum multivariate polynomials of least total degree $\mathsf{degF}$, $\mathsf{degG}$ respectively whose values are given at a set of points ...
4
votes
1answer
111 views

Stabilisers of group actions

Let $G$ be an algebraic group acting on an irreducible algebraic variety $X$ over an algebraically closed field $k$ of characteristic $0$. Suppose there exists some point $x \in X$ whose ...
1
vote
0answers
67 views

identity component of a formal group

Let $G=\operatorname{Spf} A$ be a formal group, the it is stated that the identity component $G^\circ$ (defined as $\operatorname{Spf} A_{\operatorname{fm}}$ for some open maximal ideal ...
1
vote
0answers
48 views

Why is the polynomial relating the invariants of a binary polyhedral group fixed by an overgroup?

Let $G$ be a finite subgroup of $\mathrm{SL}(2,\mathbb{C})$ and $N \triangleleft G$ a normal subgroup. Let $x, y, z$ be the fundamental invariants for the standard action of $N$ on $\mathbb{C}^2$, ...
2
votes
0answers
74 views

Dominating affine varieties over $k$ with affine smooth varieties over $k$

Given a geometrically integral affine variety $X:=\mathrm{Spec}(K[X_1,\ldots, X_n])/(f_1,\ldots, f_m)$ over a possibly imperfect field $K$, does there always exist an affine variety $\tilde{X}$ ...
7
votes
3answers
276 views

Exact sequences of groups and Tannakian formalism

By work of Deligne and others (I am following Deligne-Milne's notes which I just began to read: http://www.jmilne.org/math/xnotes/tc.pdf) we know that a given affine group scheme G can be recovered ...
0
votes
1answer
255 views

Reference for a lemma on étale maps

The Stacks Project has the following really nice Lemma concerning étale maps of rings: Let $A\rightarrow B$ be a finitely presented, étale morphism of rings. Then there exists a presentation $$ ...
6
votes
1answer
190 views

Acyclicity of the sheaf of real analytic differential forms

Let $M$ be a real analytic manifold. In the book "Sheaves on Manifolds" by Kashiwara and Schapira it is claimed on p. 127 (without reference or proof) that the Poincare lemma holds for the de Rham ...
0
votes
0answers
55 views

Restriction of motivic nearby cycles

Let $h:Y\to \mathbb C$ be a regular map and let $f:X\to \mathbb C$ be the restriction of $h$ to a closed subvariety $X\subset Y$. Both $X$ and $Y$ are assumed to be smooth. The maps $h,f$ induce ...
1
vote
1answer
97 views

ideals generated by two elements in the polynomial ring of two variables over a field

Let $k$ be a field. For example, $k=\mathbb{Q}$ or $\mathbb{Z}/p$, $p$ prime. Let $k[x,y]$ be the polynomial ring. Let $f,g\in k[x,y]$. Let the ideal $I=(f,g)$ be the ideal of $k[x,y]$ generqated ...
2
votes
1answer
141 views

Is $K^0(X)\to K_0(X)$ monomorphic for a noetherian scheme $X$?

This question is related to the MO questions What is the difference between Grothendieck groups K_0(X) vs K^0(X) on schemes? and Does a fully faithful functor between triangulated categories induce ...
0
votes
0answers
80 views

Locally free sheaves and flat families of projective scheme

Let $f:X \to Y$ be a flat proper morphism of noetherian projective schemes and $\mathcal{F}$ is a coherent sheaf on $X$. Suppose for all $y \in Y$, $\mathcal{F} \otimes_{\mathcal{O}_Y} \mathcal{O}_y$ ...
-2
votes
0answers
96 views

Proj of some ring [on hold]

Let $R=\mathbb C[x_1,x_2,x_3,x_4,x_5,y_1,y_2,y_3,y_4,y_5]$ be the polynomial ring and let $S$ be the subalgebra generated by $x_1x_2x_3x_4x_5,x_1x_2x_3x_4y_5,\cdots, y_1y_2y_3y_4y_5$ (the generating ...
2
votes
0answers
39 views

Is the Quot scheme of finite length quotients with prescribed composition factors projective?

Assume we have a scheme $X$ over a field, say $\mathbb{C}$, and a "nice" sheaf of ring $R$ on it. $E$ denotes a left $R$-module. We denote by $Q:=Quot_R(E,n)$ the scheme classifying quotients ...
2
votes
1answer
134 views

Deformation of curves and closed immersions

Let $\pi:\mathcal{C} \to B$ be a (flat) family of complex projective schemes of pure dimension $1$ with fixed Hilbert polynomial, in particular, for some $n \ge 3$, $\mathcal{C} \hookrightarrow ...
4
votes
1answer
143 views
+50

Radius of the largest enclosed ball in the convex hull of an algebraic variety

Let $\mathcal{V}\subset\mathbb{R}^n$ be a real compact algebraic variety. Let $\mathcal{V}^c$ be the convex hull of $\mathcal{V}$ and let us assume that $\mathcal{V}^c$ has nonzero n-dimensional ...
2
votes
0answers
98 views

Simply connected Kahler manifold without any effective divisor

Does anyone know an example of a simply-connected compact Kahler manifold without an effective divisor? Does anyone know a reference on this topic? Thanks!
1
vote
0answers
113 views

Vector bundle is semistable if only if it's pull back is semistable?

If $X$ is a smooth projective variety and $D$ is a divisor on $X$, and let $i:D\longrightarrow X$ be the closed immersion. Let $E$ be a vector bundle on $X$. Are there any theorems which say that $E$ ...
4
votes
2answers
186 views

Examples of surface automorphisms with no periodic points

Consider a smooth projective complex surface $S$ with an automorphism $g:S\to S$. A point $p$ is periodic if it has finite orbit under iterates of $g$. What are some examples of surface ...
2
votes
1answer
174 views

Vanishing in etale motivic cohomology

As far as vanishing is concerned, the usual motivic cohomology has the following two properties (for a smooth scheme $X$ over a field): $H^{p,q}(X, \mathbb Z) = 0$, if $p > q + dim(X)$; and ...
2
votes
1answer
173 views

Given a map of classifying spaces, can the target be described as a groupoid quotient of the source mod some action of some (co)kernel?

Let $H \to G$ be a homomorphism of affine algebraic groups (over characteristic $0$, if it matters). The case I care most about is when $H \to G$ is an inclusion. There is a corresponding map $f: ...
7
votes
1answer
200 views

how do automorphisms of elliptic curves act on the Tate module?

Let $E/k$ be an elliptic curve over some algebraically closed field $k$ of characteristic $p\ge 0$. It's known that $Aut(E)$ acts faithfully on the Tate module $T_\ell(E)$ ($\ell\ne p$) with ...
3
votes
2answers
214 views

Proof that image of a polynomial map is a cone

Consider the nonlinear mapping $\phi: \mathbb R^{2 \times 2} \to \mathbb R^3$ given by $X \mapsto \begin{pmatrix} x_{11} x_{21} \\ x_{11} x_{22} + x_{21} x_{12} \\ x_{12}x_{22} \end{pmatrix}$. I ...
-3
votes
0answers
103 views

meromorphic sections of line bundles over riemann surfaces [closed]

What is the obstruction on two holomorphic line bundles over a Riemann surface (with non-zero genus), which are associated to two divisors with the same degree, being isomorphic? In genus zero case, ...
7
votes
2answers
247 views

Fibrations of projective varieties

Let $f:X\rightarrow Y$ be a flat morphism of normal projective varieties with fibers of positive dimension (in particular all the fibers are connected and of the same dimension). Let $g:X\rightarrow ...
0
votes
0answers
78 views

Are Einstein-Hermitian connections on a stable vector bundle ever algebraic?

Let $X$ be a smooth, complex projective variety with ample line bundle $H$, and let $E$ be a poly stable vector bundle on $X$. Then there is a unique Hermitian-Einstein connection on $E$. Is this ...
3
votes
2answers
282 views

Can there be a non-trivial epimorphism (of rings) from a field? [closed]

I apologize if this question is trivial, but I just cant figure it out. Let $K$ be a field and let $K\longrightarrow A$ be an epimorphism of rings. Is it necessary that $A=K$?
4
votes
2answers
212 views

Universal curve of stacks of stable curve

Let $\overline{M}_{g,A}$ the moduli stack of pointed genus $g$ stable curves with weights $A = (a_1,...,a_n)$ introduced in Brendan Hassett, Moduli spaces of weighted pointed stable curves, Adv. ...
0
votes
0answers
60 views

Classification of line bundles by Griffiths and Harris [migrated]

I am reading pages 132 and 133 of Principles of Algebraic Geometry by Griffiths and Harris. They consider an holomorphic line bundle $L\to M$ over a manifold $M$ and an open cover $\left\{ ...
4
votes
0answers
129 views

Models for the moduli space $\overline{M}_{1,n}$

Let $\overline{M}_{1,n}$ denote the coarse moduli space of $n$-pointed elliptic curves. Is there an explicit description of these spaces (a la Kapranov's construction) for low $n$? Apparently this ...
-3
votes
0answers
71 views

a question about minimal reduction of an ideal [closed]

let $I=(x^2,y^2,xyz)$ be an ideal of $R=k[[x,y,z]]$, where $k$ is field. Then $(x^2,y^2)$ is a minimal reduction of $I$.
-3
votes
0answers
72 views

a question about basic ideal [closed]

An ideal that has no reduction other than itself is called basic. If $(R,m)$ is Noetherian local ring and $I$ is a basic ideal of $R$, then $\overline{{m}{I^n}}\cap I^n={m}I^n$
5
votes
0answers
304 views

Correspondence between line bundles and $U(1)$-bundles: a possible mistake from physicists? [closed]

I am reading a paper written by physicists and they say the following: Let $(L,h,\nabla)$ be an holomorphic Line bundle equipped with a hermitian metric $h$ and Chern connection $\nabla$. If ...
2
votes
1answer
333 views

Does this $\mathbb{Z}_p$-algebra morphism induce a closed immersion on the generic fiber?

Let $R$ be a local and smooth $\mathbb{Z}_p$-algebra and $B$ an $R$-algebra of finite type which is an integral domain with $\operatorname{dim}B\leq \operatorname{dim}R$ such that $B/(p)$ is ...
2
votes
1answer
158 views

Schemes associated to algebraic cycles and local complete intersection

We know that for an effective divisor on a smooth projective variety there is a natural way of associating to it a scheme, in particular using the Cartier divisor. Can we do the same for higher ...
0
votes
0answers
65 views

Compact locally conformal Kahler manifolds with non-zero Euler characteristic

I would like to know if there exist eight-dimensional compact manifolds such that: It has SU(4)-structure (and hence it is spin). It is locally conformal Kahler (and not Kahler). Its Euler ...
2
votes
1answer
204 views

Are schemes which agree on open set and its complement equal? - w/ applications to initial ideals/tropical basis

I appreciate the comments so far and am modifying based on something closer to the problem I'm interested in. I started out with something far too general. This is probably easy, but I have been ...
5
votes
2answers
196 views

When can stable map space have non-reduced structure?

My question is on which situation stable map space $\overline{M_{g,n}}(X,\beta)$ can have non-reduced structure. There is many example of stack structure from automorphism of stable curve but I ...
0
votes
1answer
101 views

question about divisors and its images

Let $X$ be a projective normal variety, $f: X\rightarrow Y$ a proper birational morphism to normal variety $Y$ and let $D$ be a Cartier divisor on $X$. Write $D=\Sigma D_k$, where the image by $f$ of ...
1
vote
0answers
58 views

Isomorphism of sheaves

Given a smooth projective variety $X$ and a semiample and big $\mathbb{Q}$-divisor $D$. We denote by $R:=\sum_{n\in \mathbb{Z}_{\geq 0}} H^0(X,\mathcal{O}_X(nD))$. Denote by $\tilde R(n)$ the ...
0
votes
0answers
106 views

Moduli space of holomorphic sections

Let $(L,M,\omega,\nabla)$ be an holomorphic line bundle over a Kahler manifold $(M,\omega)$ equipped with the Chern connection $\nabla$. Let $\Gamma(L)$ denote the space of holomorphic sections of ...
0
votes
0answers
66 views

Equivalence of holomorphic line bundles from Kahler potentials

Let $(M.\omega)$ be a Kahler manifold with fundamental form $\omega$. Then $\omega$ is closed and by the $\partial\bar{\partial}$-lemma on every contractible open set $U\subset M$ we can write ...
4
votes
0answers
157 views

Does integral closure commute with pushforward

Suppose that $\pi : Y \to X$ is a proper birational morphism between normal varieties (schemes, whatever). Suppose that $I$ is an ideal sheaf on $Y$. One can form $\pi_* I$ and construct an ideal ...
1
vote
0answers
212 views

Algebraic K-theory of complex varieties

Maybe this question is trivial, but I was not able to find an answer. The question is this: Consider the algebraic K-theory of smooth complex projective varieties (such that the K-theory and the ...
5
votes
1answer
231 views

Severi's theorem of base and Hilbert polynomial

Let $X$ be a smooth projective variety over $\mathbb{C}$ satisfying $H^1(\mathcal{O}_X)=0$. Fix $i:X \to \mathbb{P}^n$ a closed immersion and let $\mathcal{O}_X(1)$ be the corresponding very ample ...
0
votes
0answers
73 views

Relative canonical sheaf of an Abelian Scheme over a noetherian base

Let $A$ be an abelian scheme over a noetherian base $S$ of relative dimension $r$, i.e., $A$ is a smooth, proper group scheme over $S$ with geometrically connected fibers of dimension $r$. The ...
-1
votes
0answers
67 views

Relative line bundle along divisor $D$

Let $X$ and $B$ be a compact Kahler manifolds and $\pi:X\to B$ be a holomorphic surjective map and $D$ be a divisor in $B$ how can we define relative canonical line bundle on $B$ along a divisor $D$? ...
2
votes
0answers
133 views

Finitness of the connected components of a stack

Let $X$ be an algebaic stack over a scheme $S$, for any $S$-scheme $Y$ we can consider the groupoid $X(Y)$ of $Y$-points. Denote by $\pi_0(X(Y))$ the set of isomorphism classes of the groupoid. Are ...
-4
votes
1answer
196 views

Integral closure of an ideal [closed]

Let $r^n+a_1r^{n-1}+\cdots+a_n=0$ be an equation of integral dependence of $r$ over an ideal $I$. Does exist a finitely generated ideal $J$, such that $J\subset I$ and $a_i\in J^i$ for all ...
6
votes
1answer
296 views

Quotient of Flag varieties

Let $G=SL_3(\mathbb{C})$ and $X=G/B$ be the associated full flag variety. Fix a non-degenerate symmetric quadratic form $Q$ on $\mathbb{C}^3$. This gives an order $2$ automorphism $F_Q$ of $X$, ...