# Tagged Questions

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

**6**

votes

**2**answers

247 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

**0**answers

112 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

**0**answers

60 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

**0**answers

80 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

**3**answers

315 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

**1**answer

293 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

**1**answer

265 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 ...

**1**

vote

**0**answers

66 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

**2**answers

235 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 $k[x,y]$ ...

**2**

votes

**1**answer

192 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

**0**answers

92 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

**0**answers

51 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 ...

**4**

votes

**1**answer

162 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 ...

**5**

votes

**1**answer

244 views

### 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

**0**answers

110 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

**0**answers

126 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 $E$ ...

**5**

votes

**2**answers

214 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

**1**answer

189 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

**1**answer

188 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

**1**answer

217 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

**2**answers

231 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 ...

**9**

votes

**2**answers

267 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

**0**answers

82 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

**2**answers

289 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

**2**answers

236 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. ...

**4**

votes

**0**answers

138 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 ...

**5**

votes

**0**answers

310 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

**1**answer

350 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

**1**answer

164 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

**0**answers

72 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

**1**answer

210 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

**2**answers

202 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

**1**answer

148 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

**0**answers

61 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

**0**answers

110 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

**0**answers

76 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

**0**answers

165 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

**0**answers

225 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

**1**answer

243 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

**0**answers

94 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 ...

**2**

votes

**0**answers

184 views

### Finiteness 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 ...

**-5**

votes

**1**answer

218 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

**1**answer

307 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 $X$, ...

**1**

vote

**0**answers

114 views

### GIT quotient of variety with finite quotient singularities

Let $X$ be a variety over $\mathbb{C}$ with finite quotient singularities, i.e. every point has a Zariski-open neighbourhood isomorphic to $U/H$ where $U$ is a smooth variety and $H$ is a finite group ...

**2**

votes

**2**answers

169 views

### Connected complement manifold

I'm working on some problem in algebraic geometry. I need a reference to the following result:
Let $h\in\mathbb{N}$ with $h\geq1$ and let $F\in\mathbb{C}\left[x_{1},\ldots,x_{h}\right]$
be a non ...

**2**

votes

**1**answer

104 views

### Families of ideals with a given initial ideal

Assume a fintie set of monomials is given. Is there a way to find the family of ideals whose initial ideal (say w.r.t revlex order) is generated by that finite set? I'll appreciate any partial answer, ...

**4**

votes

**1**answer

311 views

### What are the automorphisms of $BG$?

Setup: Let's work in the category of schemes over $\mathbb C$. Let $G$ be a finite group. Let $BG=[pt/G]$ be the classifying stack of principal $G$ bundles. This is a fiberd category over the big ...

**4**

votes

**2**answers

276 views

### Find all possible rational values of a parametric quartic such that it is reducible

Description: Given the following parametric quartic polynomial
$y^4 - 28 z y^3 - 14 (656 - 328 z + 83 z^2) y^2 +
4 z (-20464 + 10232 z + 3409 z^2) y +
91 (62208 - 62208 z + 41504 z^2 - 12976 z^3 ...

**4**

votes

**1**answer

206 views

### Reference request for division algebras, over $\mathbb{Q}_{p}((t))$

I was looking for a possible reference that would answer the following question,
Let $\mathbb{Q}_{p}$ be the $p$-adic numbers and $\mathbb{Q}_{p}((t))$ be the field of Laurent polynomials over ...

**1**

vote

**0**answers

150 views

### What is the co-kernel of the morphism of vector bundles?

Let $X$ be a surface, and $i:C\subset X$ be a smooth curve. Let $A$ be a line bundle on $C$, and $E$ be a vector bundle of rank $r$ on $X$.
Suppose there is a surjection: $E\longrightarrow ...