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

**3**

votes

**1**answer

45 views

### Which weighted projective spaces (and their finite quotients) are local complete intersections?

Let $G$ be a finite subgroup of $Gl_{n+1}(k)$ (for $k$ being an algebraically closed field). My question is: do there exist examples of $G$ such that the corresponding quotient $P$ of $\mathbb{P}^n$ ...

**-2**

votes

**0**answers

35 views

### Rearrangement of difficult algebraic equations [on hold]

I have always had difficulty when rearranging large equations of functions to find the roots and also the turning points(using derivatives) and was hoping some people could give me some tips when it ...

**7**

votes

**0**answers

125 views

### Calculation-free proof of the Weyl Integral formula for U(n)

The Weyl integral formula states that if f is a class function on U(n), T is the torus of diagonal matrices in U(n), and dU(n) and dT are the standard Haar measures on U(n) and T, then
$$\int_{U(n)} ...

**1**

vote

**1**answer

87 views

### On conflicting descriptions for tor of a local cohomology group

Let $X$ be a smooth projective surface and $C$ a Cartier divisor on $X$. Denote by $\mathcal{H}^1_C(\mathcal{O}_X)$ the sheaf associated to the presheaf $U \mapsto H^1_{C \cap U}(\mathcal{O}_X|_U)$. ...

**0**

votes

**0**answers

68 views

### Is the localization sequence exact in the middle mod. algebraic equivalence?

Let $X$ be a smooth projective $k$-variety ($k=\bar k$) and $U\subset X$ an nonempty open subset. Is it true that a cycle algebraically equivalent to zero in $U$ comes from a cycle of $X\backslash U$ ...

**2**

votes

**0**answers

52 views

### space of stability condition for an elliptic curve

Let $E$ be an elliptic curve. I want to understand why $\mathrm{Stab}(E)/\mathrm{Aut}(D^b(E))$ is a $\mathbb{C}^\times$-bundle over the moduli space $\mathbb{H}/\mathrm{SL}(2,\mathbb{Z})$ of elliptic ...

**8**

votes

**0**answers

95 views

### Reduction formula for Schubert polynomials

In my endless fiddling with formulas I discovered one that fills in the blanks in a generic formula I saw in a paper, but I'm wondering if maybe it's already known and the paper was just mentioning ...

**2**

votes

**0**answers

201 views

### Weil Petersson metric on moduli space of Calabi Yau manifolds

Let $f:(X,D)\to Y$ be a holomorphic fibre space where $D$ is divisor with conic singularities and let fibres $(X_s,D_s)$ are log Calabi-Yau pair .i.e $K_X+D$ is nummerically trivial, then we have ...

**4**

votes

**1**answer

130 views

### In a noetherian commutative ring with only one associated prime, is the nilradical locally free?

The title says it all.
I suspect that the answer in general is no, although my intuition tells me that a jump in the dimension of the fibre of the nilradical at some point of Spec(A) can occur only ...

**1**

vote

**1**answer

108 views

### Can one drop the locally free assumption in projection formula on a projective bundle?

Let $X$ be a noetherian scheme over $\mathbb{C}$, and let $E$ be a locally free sheaf of finite rank over $X$. Then we have the projective bundle $f: \mathbb{P}(E)\rightarrow X$.
Now $f$ is a flat ...

**1**

vote

**0**answers

55 views

### Endomorphism of Chow goup induced by a birational map

Let $\phi:X\dashrightarrow Y$ be a birational map between smooth projective $k$-varieties ($k=\bar k$) and $\Gamma$ be the closure of the graph of $\phi$. In Fulton's intersection theory example ...

**2**

votes

**1**answer

96 views

### Looking for Schmickler-Hirzebruch' monograph on elliptic surfaces

I wonder if it is possible to find (and if yes, where?) an electronic copy of the following monograph:
Author: Schmickler-Hirzebruch, Ulrike
Title: Elliptische Flächen über $\mathbb P^1(\mathbb ...

**2**

votes

**0**answers

76 views

### Dimension of the singular locus of $\mathcal M_X(r,d)$

Let $X$ be an algebraic smooth curve of genus $g$ over $\mathbb C$, and let $\mathcal M_X(r,d)$ (resp $\mathcal M_X^0(r)$) be the moduli space of vector bundle of rank $r$ and degree $d$ (resp. with ...

**1**

vote

**0**answers

74 views

### Curve meeting an open subset

I would like a reference for the following (easy/classical?) result:
Let $X$ be a quasi-projective irreducible algebraic variety of dimension $\ge 1$, defined over an algebraically closed field $k$ ...

**5**

votes

**2**answers

178 views

### “Interesting” projective varieties being quotients of $\mathbb{A}^n\setminus \{0\}$ by an action of an algebraic group?

The algebraic (multiplicative) group $G^m$ acts on $\mathbb{A}^n$ (diagonally) and the quotient of $\mathbb{A}^n\setminus \{0\}$ by $G_m$ is $\mathbb{P}^{n-1}$ (which is a proper variety). I would ...

**4**

votes

**0**answers

156 views

### What is behind the Hodge conjecture? [duplicate]

My question is quite naive, and my knowledge limited on the subject. I heard lot of talks about Hodge conjecture. I wanted to ask about an intuitive way to figure out why we should care about Hodge ...

**1**

vote

**1**answer

147 views

### How to extend an equivariant map from a compact Lie group

Let $G$ be a compact Lie group and let $H$ be a closed subgroup of it. Let $g$ be a torsion element of $G$ and $C_G(g)$ the centralizer of it. Let $Y$ be a $C_G(g)-$space. I'm working on the space ...

**1**

vote

**0**answers

99 views

### Family $(X_y,D_y)$ with trivial canonical bundles

Let $i:D\hookrightarrow X$ and $f : X \to Y$ be holomorphic mappings of complex manifolds
such that $i$ is a closed embedding and $f$ as well as$ f \circ i$ are proper and smooth and $D$ is a divisor. ...

**5**

votes

**3**answers

234 views

### Smooth complete intersections and sharpness of the Chevalley-Warning theorem

Let $X$ be a complete intersection in $\mathbb{P}^n$ of multidegree $(d_1,\ldots,d_r)$. If we're working over a finite field $\mathbb{F}_q$, the Ax-Chevalley-Warning theorem says that if $X$ is in the ...

**1**

vote

**1**answer

70 views

### Existence of real solutions for a system of linear and quadratic equations

Suppose we have a system of linear and quadratic equations with rational coefficients and we want to find out whether this system has a real solution or not. The actual values of a solution is not ...

**0**

votes

**1**answer

267 views

### Andre-Oort for conjecture [on hold]

Is Andre-Oort conjecture expected to hold for complex analytic topology? If yes, do the recent results on abelian type Shimura varieties cover this case?
Sorry for my ignorance, but several ...

**4**

votes

**1**answer

158 views

### Existense of semi-stable vector bundles on smooth curves in positive characteristic

Let $k$ be an algebraically closed field of positive characteristic and $X$ be a smooth projective curve over $k$ of genus $g \ge 2$. Fix a polarization $L$ on $X$. Does there exist a semi-stable ...

**4**

votes

**0**answers

76 views

### Overview and/or reference of theory of pro-universal covers?

This question will contain very little in the way of concrete information, because I don't have much to go on. I've heard whispers of something called a "pro-universal cover," which is the inverse ...

**7**

votes

**2**answers

594 views

### Weil's paper under a pseudonym on deforming singular varieties

I am looking for a paper of Weil that is published under a pseudonym, in which he proves a statement along the lines of: a singular algebraic variety cannot be deformed into a nonsingular one.
Thanks ...

**-6**

votes

**0**answers

138 views

### Formal generic fibre and Fermat's Last Theorem

Set $A_{n} \colon= {\Bbb F}_p[[S_1,...,S_n]]$ and
$A_{n,d} \colon= {\Bbb F}_p[[S_1,...,S_n]][[X_1,...,X_d]]$ be a $d$-variables formal power series ring over $A_n$. We denote by $K$ the fractional ...

**10**

votes

**1**answer

184 views

### What can we say about tropical maps $\mathbb{P}^1 \to A$ for an Abelian variety $A$?

It's well known that maps $\mathbb{P}^1_\mathbb{C} \to A$ are constant for any Abelian variety $A$ (in fact, for any complex torus).
Is there any similar statement in the tropical case? Naively, the ...

**2**

votes

**0**answers

82 views

### Which sheaves on a projective bundle are flat over the base scheme?

Assume $X$ is a noetherian scheme over $\mathbb{C}$ and $E$ a locally free sheaf of finite rank on $X$. Denote the the associated projective bundle by $f: \mathbb{P}(E)\rightarrow X$.
Are there any ...

**2**

votes

**0**answers

58 views

### Inverse limits of schemes and open subsets

Let $R$ be a discrete valuation ring, $\{A_i\}_{i \in I}$ be a direct system of $R$-algebras and $A$ the limit of the system. Let $X$ be a noetherian projective scheme over $\mathrm{Spec}(R)$. ...

**2**

votes

**0**answers

38 views

### Gonality and Clifford dimension of curves on a K3 surface

Let $X$ be a K3 surface. Let $L$ be an ample line bundle on $X$. When/how can we say that any smooth curve $C\in |L|$ has maximal gonality $k=[\frac{g+3}{2}]$ and Clifford dimension 1. Is there some ...

**6**

votes

**0**answers

217 views

### Is Frac $\mathbb{Z}((x))$ Hilbertian?

Note that Frac $\mathbb{Z}((x)) \ne\mathbb{Q}((x))$.
As a result of Some questions about the ring Z((x)), we know that it is a Dedekind domain with uncountably many primes, each of which is of the ...

**2**

votes

**1**answer

201 views

### $dd^\mathbb{C}$-lemma on pair $(X,D)$

Let $X$ be a Kähler manifold with a simple normal crossing divisor $D$, i.e., pair $(X,D)$. Let $\omega$ and $\omega'$ be two Kähler forms in the same Kähler class then have we $dd^\mathbb{C}$-lemma ...

**2**

votes

**1**answer

66 views

### Veronese embeddings and locally free resolutions

Let $i : \mathbf P^1 \to \mathbf P^2$ be the second Veronese embedding. Clearly, $i_\star \mathcal O_{\mathbf P^1}$ has a locally free resolution of the form
$$
0 \to \mathcal O_{\mathbf P^2} (-2) ...

**2**

votes

**2**answers

164 views

### Finite orbits on an elliptic curve with two generic involutions

Let $C$ be a (very) general genus 1 curve embedded in $\mathbb{CP}^1\times \mathbb{CP}^1$ as a (2,2)-divisor.
Each projection defines $C$ as a double cover of $\mathbb{CP}^1$ and induces an ...

**-3**

votes

**0**answers

138 views

### Has this special functor a left and a right adjoint? [on hold]

I would like to know if there exists the left and the right adjoint functors of the functor : $ X \to \displaystyle \bigoplus_{ n \geq 0 } H^n ( X , \mathbb{Q} ) = \displaystyle \bigoplus_{ n \geq 0 } ...

**1**

vote

**0**answers

73 views

### When does $R [x]/I $ have infinitely many idempotents in special case?

At < When does $R [x]/I $ have infinitely many idempotents? >, Er_Ro asked the following question.
Let $R $ be a commutative ring with identity and $R[x] $ its polynomial ring. I am looking ...

**1**

vote

**1**answer

209 views

### Bertini-type theorem in positive characteristic [on hold]

Let $f:X \to Y$ be a morphism of finite type of irreducible schemes over an algebraically closed field of characteristic $0$. Assume that $Y$ is non-singular. Let $x \in X$ be a closed point and ...

**3**

votes

**0**answers

124 views

### A question about equivariant sheaves [on hold]

Suppose we have an G-equivariant sheav $\mathcal F$ on a smooth variety $X$. Can we split $\mathcal F$ as sum of eigensheaves? (I have seen this for structure sheaf but not sure if we can do it for ...

**0**

votes

**0**answers

133 views

### Tautological line bundle after blow-up

Let $X$ be a projective manifold, and $Z$ be a submanifold of $X$ with codimension at least 2. Let $Y$ be the blow-up of $X$ along $Z$ with the exceptional divisor $E$. Then $\pi_*:T_Y\rightarrow ...

**5**

votes

**1**answer

420 views

### Intuition behind salient numbers in number of h-cobordism classes of smooth homotopy n-spheres

The Wikipedia article on Exotic Sphere displays this sequence of numbers (see also OEIS A001676 and the Milnor link therein) for the order of the classses as
$$1, \;1, \;1,\; 1,\; 1, \;1, \;28,\; ...

**2**

votes

**1**answer

164 views

### A basic question on local cohomology

I had posted this question on stackexchange but did not get any response, hence putting it up on mathoverflow.
Let $X$ be a smooth, projective variety, $i:X \hookrightarrow \mathbb{P}^n$ a closed ...

**14**

votes

**1**answer

913 views

### Grothendieck's “La longue Marche à travers la théorie de Galois”

It seems that Grothendieck's familly has given permission for the distribution of his unpublished works, so I hope it is ok to ask this.
Is there any way to obtain a copy (online or not) of "La ...

**0**

votes

**0**answers

50 views

### Does the action of a 2-torsion line bundle on $Pic^d(C)$ fix the number of sections?

Let $C$ be a smooth projective curve over $\mathbb{C}$. Let $A$ be a degree $d$ line bundle on $C$, and $M$ be a degree 0 line bundle on $C$ such that $M^2=\mathcal{O}_C$, that is, it is a 2-torsion ...

**2**

votes

**0**answers

55 views

### What is known about topological equivalence of polynomial dynamical systems on two different domains in R^n?

The question is mainly about $\it flows$, not maps (i.e., continuous time, not discrete time).
Is it known if the study of polynomial dynamical systems on $\mathbb R^n$ can be reduced to the study ...

**8**

votes

**0**answers

199 views

### Quantum cohomology of line bundles over $\mathbb P^N$

Let $n,N$ be two positive integers. Consider the total space of the line
bundle $\mathcal O(-n)$ on $\mathbb C\mathbb P^N$. This is an algebraic variety with an action of $G=GL(N+1,\mathbb C)\times ...

**3**

votes

**1**answer

170 views

### Veronese embeddings of elliptic curves in weighted projective space

Let $E$ be an elliptic curve and $D_k=kp$ a divisor on $E$, where $p\in E$, for $k\in\mathbb{N}$.
Then we can reconstruct $E$ from the graded ring $R(D_k)=\bigoplus_{n\geqslant0}\mathcal{L}({nD_k})$: ...

**3**

votes

**1**answer

93 views

### Condition on moment polytope for a toric manifold to be Fano

Suppose $M$ is a symplectic toric manifold. This means there is a compact torus
$T$ that has a Hamiltonian action on $M$, with moment map $\mu:M \to \mathfrak t^*$, and $\dim(M)=2\dim(T)$. Can one ...

**5**

votes

**0**answers

85 views

### model theory of non-reduced schemes

In model theory one studies Boolean algebras of definable sets of complete theories. For many theories definable sets are in direct correspondence with geometric objects, for example, definable sets ...

**6**

votes

**2**answers

730 views

### What is the role of projective spaces in GAGA?

The GAGA theorem is a celebrated elaboration of the idea that complex analytic and complex algebraic geometry are equivalent, at least for smooth projective varieties/manifolds.
I am aware why this ...

**1**

vote

**0**answers

77 views

### A Nodal curve embedded in a smooth variety, is always regularly embedded?

I will consider more simple situation. Let C be a nodal curve which is a union of two $\mathbb{P}^1$, $C_1,C_2$. Which meets at a node $p$.
Consider $C$ is embedded in a smooth variety $Y$. Assume ...

**-4**

votes

**0**answers

82 views

### how to prove that $k[XY]/(XY)$ is a $k[Z]-$ module finitely generated? [closed]

I'm wonrking in this question: Let $A=k[X,Y]/(XY)$, and take the homomorphism $k[Z]\rightarrow A$ given by $Z\to X+Y$. Show that this homomorphism is finite and injective.
For the injectivity I can ...