# Tagged Questions

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

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votes

**1**answer

314 views

### why quintics are Calabi-Yau?

Why quintics are Calabi-Yau? Is there a explicit formula of the holomorphic volume form?

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**0**answers

115 views

### singularities and zeros of series and reverse problem [closed]

Given a series $$S=\sum_{n=1}^{\infty}a_n x^n,a_n\in \mathbb{N}$$ we know that it may have zeros,poles,branch points or natural boundary.Reversely,given it's zeros,poles,branch points or natural ...

**6**

votes

**3**answers

269 views

### Weak Fano and Log fano varieties

A projective smooth variety $X$ is weak Fano if $-K_X$ is nef and big. We say that $X$ is log Fano is there exists a divisor $D$ such that $-(K_X+D)$ is ample and $(X,D)$ is Kawamata log terminal.
Is ...

**3**

votes

**1**answer

185 views

### Surfaces singular along a curve

Let $C\subset\mathbb{P}^3$ be a smooth curve a degree $d$ and genus $g$. Let $\mathcal{S}$ be the system of surfaces of degree $k$ in $\mathbb{P}^3$ containing $C$ with multiplicity $\beta$.
What is ...

**2**

votes

**1**answer

341 views

### Two questions about the grassmannian

There are two statements about the grassmannian (of complex k-planes in n-space embedded via Plucker coordinates) that I have encountered in several places never accompanied with a proof or reference.
...

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votes

**1**answer

189 views

### Cotangent bundle of coadjoint orbit is stein manifold?

Let me first define stein manifolds and coadjoint orbits.
A complex manifold $X$ of complex dimension $n$ is called a Stein manifold if the following conditions hold:
$X$ is holomorphically convex, ...

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**0**answers

77 views

### Aysmptotic comparison of L^2 sections versus generating sections

Let $s_1,\ldots, s_k$ be linearly independent global holomorphic sections of a holomorphic line bundle $E$ over a compact algebraic manifold $X$, with volume form $\Omega$.
For $m$ large, let ...

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**0**answers

36 views

### connected components of a real algebraic variety and its hyperplane section

Let $X$ be a smooth projective variety of dimension at least $2$ over the real numbers $\mathbb{R}$ and $H \subset X$ a smooth hyperplane section. Assume that the set of real points is non-empty for ...

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votes

**1**answer

337 views

### Topological degree and polynomial degree

Let $F:\mathbb{C}^n\to \mathbb{C}^n$ be a homeomorphism homogeneous of degree 1 (i.e., $F(tx)=tF(x)$, $t>0$) and $g:\mathbb{C}^n\to \mathbb{C}$ a homogeneous polynomial of degree $k$. Let $L$ ...

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votes

**1**answer

330 views

### On a question motivating Lurie's treatment of formal moduli problems

Lurie, in his ICM 2010 proceedings paper Moduli Problems for Ring Spectra (pdf), says that one motivating problem (for him, I presume, and possibly others) for thinking about formal moduli problems is ...

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vote

**1**answer

121 views

### Universal coefficient theorem for local ring

Let $R$ be a commutative local artin $k$-algebra,where $k$ is a field with characteristic $0$.I wonder whether universal coefficient theorem holds in this case.Namely,if $C$ is a chain of flat ...

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**2**answers

169 views

### When is a blow-up of $\mathbb{P}^2$ a Mori Dream Space?

Let $p_1,...,p_k\in\mathbb{P}^2$ be general points. Let us consider the blow-up $X_k = Bl_{p_1,...,p_k}\mathbb{P}^2$. It is clear that if $k\leq 3$ then $X_k$ is toric and hence a Mori Dream Space. ...

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154 views

### Nonsingular is rational singular

I know the following result is well known to the experts, but what are the good references or proofs:
Let $X,Y$ be a smooth variety with projective, birational morphism $f: X \to Y$. Let $L$ be a ...

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60 views

### Compactification of the affine space with a del Pezzo surface

Is there some nice compactification of the affine space with a del Pezzo surface of degree $\le 4$ (for example with a cubic surface) ?
More precisely, I would like a projective algebraic variety $X$ ...

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**0**answers

113 views

### integral hard Lefschetz

I am looking for examples $(X,\eta)$ where the integral hard Lefschetz is an isomorphism:
$X/k$ is a smooth projective variety of dimension $d$ over an algebraic closure of a finite field and $\eta ...

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votes

**2**answers

261 views

### An affine singular surface

Let $n$ be a positive integer and let $A$ be the subring of ${\mathbb C}[x,y]$ generated
by $x,xy,...,xy^n$. Let $S=Spec(A)$. This is an affine surface, which is clearly singular if
$n\neq 1$. Is ...

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**2**answers

137 views

### Examples for curve not 1-connected but $h^0(C, O_C)=1$

Suppose $X$ is a surface, are there examples for curves on X which not 1-connected, but $h^0(C,O_C)=1$? (Here a curve is an effective divisor, not necessarily reduced or irreducible) (1-connectedness ...

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**1**answer

210 views

### Non-cohomological proof that a noetherian scheme $X$ is affine if its reduction $X_{red}$ is affine

Can we prove that a noetherian scheme $X$ is affine if its reduction $X_{red}$ is affine without using cohomology?
Remark
Here is a similar question.

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votes

**6**answers

1k views

### Down-to-earth expositions of Hodge theory

What are nice expositions of Hodge theory not using advanced language of algebraic geometry or category theory?
Of course, since I haven't found a (for me) readable introduction, I don't know what I ...

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votes

**1**answer

73 views

### Computing the minimal free resolution of a coherent sheaf on projective space

Most books on commutative algebra explain Grobner bases in the non graded case and minimal free resolutions in the local case. I like projective geometry and want to compute the minimal free ...

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**0**answers

158 views

### Solving $P=AB,Q=BA$, in the unknowns $A,B$

Let $p\geq q$ $P\in M_p(\mathbb{C}),Q\in M_q(\mathbb{C})$. We seek $A\in M_{p,q},B\in M_{q,p}$ s.t. $P=AB,Q=BA$. The NS conditions for the existence of $(A,B)$ are given in
On the matrices AB and BA. ...

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**0**answers

50 views

### Homomorphisms between groups of Hermitian type and Hodge type and orthogonality

By a group of Hermitian type I mean a real group $H$ with a maximal compact subgroup $K$ such that $H/K$ is Hermitan Symmetric domain. A real group $W$ is called of Hodge type if the associatd ...

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**1**answer

112 views

### Arithmetic property of a surface of general type

In my previous post I asked about the hyperbolicity of the affine surface $S'=\{zw \neq o\}$ in the projective surface $z^2 = P(x) Q(y)$ in $\mathbb{P}^3$, where $P$ and $Q$ are two general ...

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**2**answers

206 views

### Etale covers of a hyperelliptic curve

Let $X$ be a hyperelliptic curve of genus at least two.
Let $Y\to X$ be a finite etale morphism with $Y$ connected.Then $Y$ is a smooth projective connected curve.
Is $Y$ hyperelliptic?
More ...

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**2**answers

1k views

### Is every Noetherian Commutative Ring a quotient of a Noetherian Domain?

This was an interesting question posed to me by a friend who is very interested in commutative algebra. It also has some nice geometric motivation.
The question is in two parts. The first, as stated ...

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**0**answers

122 views

### Are there perverse sheaves on abelian varieties with small Euler characteristic?

Let $A$ be a simple abelian variety of dimension $g$. Let $K$ be an irreducible perverse sheaf on $A$. We know that $\chi(A,K)\geq 0$. (Corollary 1.4 of Franecki and Kapranov.) How small can ...

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**0**answers

277 views

### Is it possible to prove Mordell's conjecture geometrically?

Let $X/k$ be a smooth curve of genus $g>1$ over a number field $k$.
By the Faltings theorem (nee Mordell's conjecture), the set of $k$ - rational points
$X(k)$ is finite. Due to the Mordell-Weil ...

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**4**answers

1k views

### Quillen's motivation of higher algebraic K-theory

Almost the same question was already asked on MO Motivation for algebraic K-theory?
However, to my taste, the answers there consider the subject from a more modern point of view.
When I open a book ...

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**1**answer

134 views

### A family of examples of (Brody) hyperbolic surfaces

Let $P(x)$ and $Q(x)$ be two general polynomial of the same degree $d$ (d can be arbitrary large). Consider the surface $S : z^2 = P(x) Q(y)$ in the projective space $\mathbb{P}^3$. I want to prove ...

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**0**answers

88 views

### Exact sequence of the fundamental group of the general fiber

Let $f\colon X\rightarrow Y$ be a morphism of complex algebraic varieties.
Let $y\in Y$ be a general point, then we have a sequence of homomorphisms
of fundamental groups induced by the inclusion of ...

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votes

**1**answer

156 views

### Kernel of a 3-isogeny between two elliptic curves

Suppose $E_1$ and $E_2$ are two elliptic curves defined over $\mathbb{Q}$ and there exists a 3-isogeny $\varphi$: $E_1 \longrightarrow E_2$. If $E_1$ has no $\mathbb{Q}$-rational point of order 3, ...

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**1**answer

199 views

### What is the relation between KC and height of rational number?

Roughly speaking,Kolmogorov Complexity of a bits string or a description is the minimal length of programs outputing a bits string,and height of rational number is logarithm of the largest numerator ...

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**2**answers

164 views

### Rank vanishing in tensor categories

Let $\mathcal{C}$ be a cocomplete $\mathbb{Q}$-linear symmetric monoidal category (if convenient, assume that it is locally finitely presentable and/or abelian). Recall that the rank (or dimension) of ...

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**1**answer

194 views

### Why should noncommutative CYs be dgas?

For a (commutative) CY manifold of dimension $n$, Serre duality implies that there are ifunctorial isomorphisms
$$\operatorname{Hom}_{D^b(X)}(E,F)\cong\operatorname{Hom}_{D^b(X)}(F,E[n])^*$$
in the ...

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**0**answers

244 views

### The logarithm over $\mathbb F_1$

In 'Cyclotomy and analytic geometry over F1', Manin proposes a version of the notion of `analytic function' over the 'field with one element $\mathbb F_1$'.
Question 1: can somebody explain or give ...

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**0**answers

60 views

### Singularities of a Schubert variety in an affine flag variety

The Iwahori orbits in the affine flag variety of an adjoint group $G$ are numbered by the extended affine Weyl group $W_e=W\ltimes X$ (where $X$ is the coweight lattice, and $W$ is the finite Weyl ...

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**0**answers

45 views

### Is the singularity of a secant variety in codimension 2 Kleinian?

Let $X\subset{\mathbb P}^d$ be a rational normal curve, $d\gg0$. Then the $k$-th secant variety $Sec^k(X)$ is smooth away from $Sec^{k-1}(X)$, and I wonder what is the singularity of $Sec^k(X)$ at a ...

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**0**answers

79 views

### Relation between dimension of Proj(S) and dimension of S

Let $S$ be Noetherian standard ${\mathbb{N}}^r$ graded ring where $S_{\underline{0}}$ is an Aritinian local ring.
$$Proj(S)=\lbrace{P\in Spec S | S_{++}\not\subseteq P, P\hspace{0.1cm} ...

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**0**answers

317 views

### A quotient stack question

Let $X$ be a proper Deligne-Mumford stack, whose normalization, $X'$, is a global quotient stack (that is, a stack of the form [W/GL_n],where W is an algebraic space) with a projective scheme as a ...

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**3**answers

458 views

### Non existence of cyclic infinite linear algebraic groups

Let $G$ be a linear algebraic group defined over some algebraically closed field $\mathbb{K}$ and also over some subfield $k\subset \mathbb{K}$. There is thus a natural group structure on the set of ...

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65 views

### Belyi's theorem for surfaces. Does exist another generalization?

Is there a generalization of Belyi's theorem in dimension $2$ (or higher) other than this by Gabino Gonzalez Diez? I haven't found no other references on the argument.
Thanks in advance

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356 views

### The link between the subfactors and the motives as enriched Galois theories?

On 29th March 2007, at the "École normale supérieure" of Paris, the mathematician Vaughan Jones, gave a conference (in French) entitled "Les sous-facteurs : une théorie de Galois enrichie" (see here), ...

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**1**answer

141 views

### Recognizing a Mukai flop

Let us first review the usual construction of a Mukai flop. Suppose $M$ is a smooth $2m$-dimensional projective variety over $\mathbb C$ containing a closed $m$-dimensional subvariety $W$, and suppose ...

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**1**answer

214 views

### Is $G_{\operatorname{red}}$ normal in $G$?

Let $G$ be an affine group scheme of finite type over a field $k$. It is well known that the associated reduced subscheme $G_{\operatorname{red}}$ of $G$ is a subgroup if $k$ is perfect. So let us ...

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vote

**2**answers

186 views

### multi-tangent space for algebraic curves

For a general plane curve of degree $\ge 4$, the number of bitangent lines is known. Also, I found that the number of tritangent planes have been worked out for some space curves given by intersection ...

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vote

**1**answer

115 views

### constructible set and fibre product

Let $P$ be certain property. Let $S \subset \mathbb{C}^n\times \mathbb{C}^m$ be a set of closed points such that for any point in $S$, it satisfies the property $P$. I know for any $x \in ...

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**2**answers

349 views

### Riemann-Roch without denominators?

The Riemann-Roch Theorem, the Grothendieck-Riemann-Roch Theorem , the Grothendieck-Hirzebruch-Riemann-Roch Theorem , all of them are well explained at Wikipedia .I would like to understand the meaning ...

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**0**answers

116 views

### Hodge Decompositions and Gamma Factors of Hasse--Weil L-Functions

Let $X$ be a projective variety over a number field $K$. If $\mathfrak{p}\unlhd\mathcal{O}_K$ is a prime ideal we can regard the Euler factor of its $m$th Hasse-Weil $L$-Function at $\mathfrak{p}$ as ...

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**1**answer

65 views

### Point-Hyperplane incidence in finite projective spaces

Let $P$ be a finite projective space of order $q$ and dimension $d$. I am interested in finding the least $k$ such that for any set $S$ of $k$ points of $P$, and for any set $S'$ of $k$ hyperplanes of ...

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**0**answers

105 views

### Is there any advantage to knowing that Gauss-Manin is Hermitian flat?

Let $S$ be a complex manifold and let $p : E \to S$ be a holomorphic vector bundle. Is there any advantage to knowing that $E$ carries a flat Hermitian metric $h$, ie a smooth Hermitian metric with ...