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

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132 views

### Blow up along a section of a smooth morphism

Let $C$ be a ground locally notherian and quasiprojective scheme. Let $\pi:S\to B$ be a $C$-morphism of finite type $C$-schemes. We call $\pi$ a $C$-smooth morphism if the morphisms $S\to C$ and $B\to ...

**2**

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**0**answers

65 views

### The Geometric Intuition behind Minimal Primary Decompositions versus an arbitrary Primary Decomposition

Given an ideal in a Noetherian domain, it has a primary decomposition. This decompositions may not be minimal. I am interested in the relationship of primary ideals that are not in a minimal primary ...

**3**

votes

**1**answer

110 views

### extending homomorphisms to semidirect products

Let $A = F_2$ be the free group on two generators (not sure if this is important.)
Suppose you have a semidirect product $A\rtimes C$ coming from some homomorphism $\beta: C\rightarrow ...

**5**

votes

**0**answers

60 views

### Constructing the normal sheaf for the plucker embedding in MAGMA (or a similar programming language)

How would one construct the normal sheaf $N_{G(2,6)/\mathbb P^{14}}$ to the plucker embedding of the grassmannian $G(2,6) \rightarrow \mathbb P^{14}$ as a sheaf in MAGMA (or another programming ...

**1**

vote

**1**answer

70 views

### how to write down comatrix of the exceptional Jordan algebra

Suppose we have exceptional Jordan algebra, which is a $3\times3$ matrix ...

**2**

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**0**answers

145 views

### Moving lemma on Abelian varieties

Suppose $Y,Z$ be two cycles on a quasi-projective manifold $X$, Chow's moving lemma says there is a cycle $Z'$ rationally equivalent to $Z$ such that $Y,Z'$ intersect properly.
Suppose $X$ is an ...

**2**

votes

**1**answer

183 views

### On surfaces with $p_g=0$, $q=1$, and $K^2=-3$ [closed]

I am having a trouble in understanding the Example 4.7 (pages 65-66), the genus two fibrations with $p_g=0$, $q=1$, and $K^2 = -3$, in "Surfaces fibrées en courbes de genre deux", Lecture Notes in ...

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125 views

### Restriction of the Canonical Divisor $K_X$ to a general fiber

Let $\ f:X\to Z$ be a surjective morphism between two smooth projective varieties with connected fibers $(f_*\mathcal{O}_X=\mathcal{O}_X)$. Let $F$ be a general fiber of $f$ and $\mbox{dim } ...

**2**

votes

**2**answers

103 views

### Biregular maps between hypersurfaces of the same degree

Let $n\geq 2$ and $\mathbb{P}^n(\mathbf{C})$ be the complexe projective space of dimension $n$. Let $H\subseteq \mathbb{P}^n(\mathbf{C})$ be a hypersurface of degree $d$ where the coordinates in ...

**0**

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**0**answers

95 views

### Cycle with integral coefficients from cycle with $\mathbb Z_l$-coefficients

Let $X$ be a $n$-dimensional ($n>2$) smooth projective variety over $k=\bar k$ of positive characteristic. Take a divisor $D\in Pic(X).$ Suppose we know that $\frac{[D]^2}{2}\in H^4(X,\mathbb ...

**5**

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**1**answer

171 views

### A conjecture associated with n-gon cut curve of degree m

This conjecture is a generalization of A theorem for cubic-A generalization of Carnot theorem, in MSE question. I'm an electrical engineer, I am not a mathematician. I don't know how to prove this ...

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**2**answers

374 views

### What does taking the graded algebra do to the Grothendieck group, and its relation to the Chow ring?

Let $X$ be a nonsingular variety. (Perhaps some/all of this works over more general smooth schemes, but let's stick to the simple case.)
In, e.g., Fulton's Intersection Theory chapter 15, and Soule's ...

**10**

votes

**1**answer

202 views

### Euler Characteristic of Real Algebraic Surfaces

Given a (compact if needed) real smooth surface $V(f)$ defined by $f\in \mathbb{R}[X,Y,Z]$, in particular it is oriented. Is there a formula which gives the Euler character of $V(f)$ ?
Thanks.

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**0**answers

73 views

### Finding generators of symmetric cones

I have a bunch of vectors $\mathbf v_i$ in $\mathbb R^n$. I would like to consider the cone $C$ spanned by these vectors, together with all the other vectors that can be obtained by permuting the ...

**2**

votes

**1**answer

227 views

### On quasi-algebraically closed fields

By Lang's theorem, a complete valued field which is the fraction field of a discrete valuation ring with an algebraically closed residue field is quasi-algebraically closed (or $C_1$).
How much is ...

**19**

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**0**answers

391 views

### Finding the octonionic analog of the K3 surface, via (almost) hyperkahler geometry?

The K3 manifold is an amazing object in mathematics which plays an important role in several fields ranging from the study of smooth 4-manifolds to algebraic geometry to differential geometry and ...

**2**

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**0**answers

30 views

### cayley transformation of bounded symmetric domain

Can anyone write down the biholomorphic map between classical bounded symmetric domains(defiend by matrixs) with their siegel upperhalf plane models. I know if it's type 2, i.e $I-Z\bar{Z}^{t}>0$ ...

**3**

votes

**1**answer

192 views

### Global sections of D-module tensor product

Let $X=\mathbb{C}^n$ be affine n-space (with the Zariski topology), $\mathcal{D}$ its sheaf of differential operators. Let $D$ be the $n$th Weyl algebra, $M$ a right $D$-module, and $N$ a left ...

**2**

votes

**0**answers

119 views

### Is the Jacobian of curve self-dual?

Given $C$ an algebraic curve, its Jacobian is isomorphic to its Albanese variety by Abel-Jacobi Theorem. But generally Jacobian and Albanese varieties are dual abelian varieties, does this imply that ...

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**1**answer

264 views

### Moduli space of boundary maps with prescribed chain and homology groups?

Let $R$ be a reasonable ring (maybe I mean a PID, or $\mathbb{Z}$, and when sufficiently desperate, a field). Now consider fixed sequences $C_n$ and $H_n$ of $R$-modules, which are tame in every ...

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**0**answers

159 views

### Kawamata's paper “A remark on the abundance conjecture”

Kawamata has posted on ArXiv a paper "A remark on the abundance conjecture", claiming to prove that "the existence conjecture of minimal models in dimension n and the abundance conjecture in dimension ...

**3**

votes

**1**answer

165 views

### Reference: Betti Numbers of the free loop space are finite

let $M$ be a compact, simply connected Riemannian manifold with dimension $< \infty$. I'm looking for a reference that
$$ \dim H_k(\Lambda M, \mathbb{Z}) < \infty, $$
is true in that case. Here ...

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votes

**0**answers

110 views

### Do complex schemes locally deformation retract onto closed subschemes in the analytic topology?

Let $X$ be a scheme of finite type over $\mathbb{C}$ and let $Z \hookrightarrow X$ be a closed subscheme. Consider the associated closed inclusion $Z_{an} \hookrightarrow X_{an}$ between their ...

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votes

**3**answers

268 views

### Existence of a morphism between two toric varieties

Does there exist a morphism between the blow-up of $\mathbb{P}^3$ in four general points and $\mathbb{P}^1\times\mathbb{P}^1$? If not why?

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130 views

### Laurent and power series over the field with one element?

Question. Is there a suitable notion of the Laurent series ring $\mathbb{F}_1((t))$ and power series ring $\mathbb{F}_1[[t]]$ in some framework for the field with one element $\mathbb{F}_1$?
For ...

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**0**answers

62 views

### critiria to prove that a morphism is an embedding

I have a forgetful map between moduli spaces, I want to prove that it is an embedding, In fact, I have a reductive algebric group (which is not constant) over a curve $X$ whose geniric fiber is ...

**3**

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**0**answers

80 views

### Can we extend a homotopy invertible chain morphisms between complexes of sheaves from a closed subspace to the whole space?

Let $X$ be a (say, topological) space and $i: Z\hookrightarrow X$ be a closed subspace. Let $Sh(X)$ and $Sh(Z)$ denote the categories of sheaves of abelian groups on $X$ and $Z$ respectively. ...

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**0**answers

97 views

### Unique solution for a specific system of polynomial equations

Let $x^{(1)}, \dots, x^{(N)} \in \mathbb R^n$ be given. Suppose all we know about the $x^{(i)}$ are the values $c_1, \dots, c_N$ as given below
\begin{align*}
&c_1 := \sum_{i=1}^N \langle v^{(j)}, ...

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**1**answer

281 views

### Hodge to de Rham spectal sequence with twisted coefficients

Let $M$ be a smooth compact Kahler manifold and let $\mathcal{F}$ be a local system on $M$.
Question 1: I assume that there exists a twisted Hodge to de Rham spectral sequence converging to ...

**5**

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**0**answers

153 views

### A relative version of Hensel's lemma?

Let $k$ be a $p$-adic field with integer ring $\mathcal{O}_k \subseteq k$, maximal ideal $m_k \subseteq \mathcal{O}_k$ and residue field $\mathbb{F}_q = \mathcal{O}_k/m_k$. Let $X$ be a smooth, ...

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votes

**1**answer

192 views

### Can you functorially “reconstruct” a branched cover of curves from its etale locus?

I'm sure this must be covered somewhere, but all the references I have only treat this in very special cases (mostly when working over fields).
Suppose $f : X\rightarrow S$ is smooth of finite ...

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**0**answers

180 views

### (very) basic question about coverings in algebraic geometry

Let $U$ be a scheme. Let $\{\phi_i:U_i \rightarrow U\}_i$ be a covering (in any topology).
Standard references say this means (in particular) that
$$\bigcup_i \phi_i(U_i) = U$$
In most cases ...

**7**

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**0**answers

174 views

### Galois group for flexes of a plane $d - ic$

Here is the first problem considered in Harris's 1979 paper, Galois Groups of Enumerative Problems (see here for the paper itself).
The Galois groups are: for the flexes of a plane $d$-ic, the ...

**1**

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**0**answers

94 views

### Representablility of fiber product over stacks

Let $(\mathcal{C},\mathcal{T})$ be a site and $\mathcal{X}$ a stack over it. Suppose $(\mathcal{C},\mathcal{T})$ subcanonical and hence for every object $U$ in $\mathcal{C}$ consider the associated ...

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**0**answers

49 views

### Intersection of affine subspaces with specific Veronese variety

Consider the mapping
$$ \Psi: \mathbb R^2 \times \mathbb R^2 \to \mathbb R^5, \\
\Psi(x^{(1)},x^{(2)}) = \begin{pmatrix} x_1^{(1)} + x_1^{(2)} \\ x_2^{(1)} + x_2^{(2)} \\ (x_1^{(1)})^2 + ...

**5**

votes

**1**answer

130 views

### Does a rational function extend to a point if all its 1-parameter extensions at that point agree?

Let $f: X \dashrightarrow Y$ be a rational map where $X$ and $Y$ are varieties over $\mathbb{C}$ with $X$ smooth and $Y$ proper (or projective if necessary). Choose an open set $U \subset X$ on which ...

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**0**answers

119 views

### When is the $I$- adic completion a faithfully flat ring extension?

Let $I$ be an ideal of a commutative ring with identity. It is well-known that the $I$-adic completion of $R$ when it is a Noetherian ring is a faithfully flat ring extension. Does this remain true if ...

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votes

**1**answer

307 views

### Local cohomology groups and linearity

I am reading local cohomology and am confused on a silly point. Let $U$ be an affine, non-singular variety and $Z \subset U$ a hypersurface section on $U$ (i.e., complete intersection in $U$ of ...

**2**

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**0**answers

76 views

### Is there an algorithm to compute the intersection of tautological classes on the moduli space of genus one curves?

Let $\overline{M}_{1,1}(\mathbb{P}^2, d) $ be the moduli space of degree
$d$ genus one curves on $\mathbb{P}^2$ with one marked point. Let
$L\longrightarrow \overline{M}_{1,1}(\mathbb{P}^2, d) $ ...

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245 views

### Counting abelian varieties over finite fields in a given isogeny class

Let $f(x) \in \mathbb Z[x]$ be a monic polynomial of degree $g$ with all roots having absolute value $\sqrt{q}$. How many principally polarized abelian varieties over $\mathbb F_q$ have $f(x)$ as the ...

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votes

**2**answers

186 views

### Birational morphism and first cohomology group

Let $f:X\rightarrow Y$ be a proper birational morphism of smooth projective varieties over $k=\bar k$ ($char(k)>0$). Is it true that $H^1(Y,\mathbb Z_l)\stackrel{f^*}\simeq H^1(X,\mathbb Z_l)$ ...

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votes

**0**answers

56 views

### Stability of bundles under blow up

Let $X$ be a surface with finitely many nodes, and $Y$ be its desingularization. Let $L$ be an ample line bundle on $X$. But the pullback $L'$ will not be ample on $Y$ (because restricted to ...

**2**

votes

**1**answer

136 views

### Higher direct image of locally constant torsion sheaf (étale cohomology)

Let $\phi:X\rightarrow Y$ be a generically smooth projective surjective morphism of algebraic varieties over $k=\bar k.$ Is it possible for $R^1\phi_*(\mathbb Z/l)$ to be supported on a divisor of $Y$ ...

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**0**answers

50 views

### kahler einstein metric for exceptional compact type hermitian symmetric space

Can anyone write down the kahler einstein metric for exceptional compact type hermitian symmetric spaces($\frac{E_6}{SO(10)*SO(2)}$ and $\frac{E_7}{E_6*SO(2)}$). I can find the bergmann kernel for ...

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**2**answers

342 views

### Twists of cubic threefolds

Let k be a field of characteristic $0$.
Let $X$ be a variety over k which is isomorphic to a smooth cubic threefold over $\bar{k}$. Then is $X$ isomorphic to a smooth cubic threefold over ...

**2**

votes

**1**answer

98 views

### Minimal model (resolution) for a specific colored operad

We know that for the operad $As:=\mathcal{F}(\mu)/(\mu\circ_1\mu-\mu\circ_2\mu)$, its minimal model is the free operad $\mathcal{F}(E)$ where $E=\mathbb{k}<\mu_2,\mu_3,\dots,\mu_n,\dots>$ is the ...

**5**

votes

**1**answer

235 views

### Why does $GL_n(F)\backslash GL_n(\mathbb{A}_F)/\prod_xGL_n(O_x)$ classify vector bundles over $X$?

Suppose $X$ is an algebraic curve, $F$ its function field, $\mathbb{A}_F$ its adele, $O_x$ the ring of integers at local field of $x$. Why does $GL_n(F)\backslash GL_n(\mathbb{A}_F)/\prod_xGL_n(O_x)$ ...

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**0**answers

93 views

### On triviality and numerical triviality of (classes of) divisors

Let $X$ be a smooth irreducible threefold, and let $H$ be an ample divisor on $X$.
Assume that $D$ is a divisor on $X$ such that $D\cdot H^2=D^2\cdot H=D^3=0$.
Question 1: Is $D$ numerically ...

**7**

votes

**1**answer

316 views

### Homotopy types of schemes

Let $X$ be a scheme over $\mathbb{C}$.
When does the topological space $X\left(\mathbb{C}\right)$ of $\mathbb{C}$-points have the homotopy type of a finite CW-complex?
When does the topological ...

**3**

votes

**3**answers

308 views

### Does there exist a holomorphic fibration of genus two over $\mathbb{P}^{1}$ with $7$ nodal singularities?

This is a problem about the holomorphic fibration on a complex manifold.
Does there exist a holomorphic fibration of genus two over $\mathbb{CP}^{1}$ with 7 nodal singularities?
If you are aware of ...