# Tagged Questions

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

**0**

votes

**0**answers

14 views

### Relative Proj and generation of sections

Let $π\colon \mathrm{Proj}_B \mathcal{A} \rightarrow B$ be a morphism constructed from a coherent graded sheaf of $\mathcal{O}_B$-algebras.
I am looking for (minimal) hypotheses for the natural ...

**0**

votes

**0**answers

58 views

### Groupes fondamentaux de Tate mixte [on hold]

Have anyone ever read deligne and Goncharov's paper,Could you give me some idea why the formula(5.16.1) is true?this paper is very easy to find on the internet.So please forgive me not giving the ...

**0**

votes

**0**answers

33 views

### Will (general points + small number of arbitrary points) impose independent condtions on plane curves?

It is well known that imposing vanishing at general points of $\mathbb P^2$ gives independent conditions on curves of degree $d$. Also, it is known a small number (something like $d+1$) points always ...

**4**

votes

**1**answer

140 views

### Must an algebraic variety with trivial tangent bundle be an abelian variety?

Suppose $X$ is an algebraic variety with trivial tangent bundle $T_X$ (not only canonical bundle $K_X$), is it true that $X$ is an abelian variety? (For the complex manifold case this is not true due ...

**1**

vote

**0**answers

79 views

### Why write GRR with the relative tangent sheaf?

The first published version of the Grothendieck-Riemann-Roch theorem, GRR for short, was written in the form
$$
\operatorname{ch}(f_!\alpha).\operatorname{Td}(Y)
=
...

**1**

vote

**1**answer

113 views

### A question on the cohomology of elliptic curves over local fields

Let $K$ be a number field,$\nu$ a nonarchimedian prime of $K$, $K_{\nu} $ the completion of $K $ at $\nu $ with maximal unramified extension $K_{\nu}^{unr} $. Let $E $ be an elliptic curve defined ...

**2**

votes

**1**answer

142 views

### 'Stalk' of vanishing cycles at $k$-point

I have a simple question on notation.
Let $S$ be a Henselian trait with closed point $s$ (with finite residue field $k$) and generic point $\eta$. Let $X/S$ be a variety. Then, we have the functor
...

**10**

votes

**1**answer

278 views

### Is there a unique commutative group structure on $\mathbb{G}_m$?

Let $S$ be a scheme and let $X := \mathrm{Spec}(\mathscr{O}_S[t, t^{-1}])$ be the underlying $S$-scheme of the $S$-group scheme $(\mathbb{G}_m)_S$. Is there only one structure of a commutative ...

**0**

votes

**0**answers

94 views

### Polynomial approximation on affine varieties [migrated]

Let $V,W \subseteq \mathbb{A}^n$ be two affine varieties over an algebraically closed field $k$ of characteristic zero and let $a,b\in k$.
Q: Can we find a polynomial $f \in k[X_1,...,X_n]$ such ...

**1**

vote

**0**answers

224 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 $\mathsf{degF}$, $\mathsf{degG}$ such that, given unequal $a,b\in\Bbb R$,
$$\mathsf{F(p)}=a, ...

**0**

votes

**1**answer

105 views

### Why does this vector bundle on the surface sit in this exact sequence?

Let $X$ be a K3 surface. Let $E$ be a semistable rank 3 vector bundle. Now suppose $0 = E_0\subset E_1\cdots\subset E_s=E$ be the Harder-Narasimhan filtration. Suppose $E_1$ is $\mu$-stable and rank ...

**5**

votes

**2**answers

220 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

101 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

56 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

79 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

296 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

273 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

232 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

64 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

150 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

**1**answer

156 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

86 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

104 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

**0**answers

41 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

138 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

**1**answer

168 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

**0**answers

100 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

118 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

**2**answers

189 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

177 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

176 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

205 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

218 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

**0**answers

104 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

**2**answers

249 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

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

**2**answers

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

**2**answers

213 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

**0**answers

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

**0**answers

130 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

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

**1**answer

334 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

159 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

67 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

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

**2**answers

198 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

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

**0**answers

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

**0**answers

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

**0**answers

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
...