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

**7**

votes

**2**answers

480 views

### Adjoining torsion points from abelian varieties

Let $L/\mathbb{Q}$ be the field generated over $\mathbb{Q}$ by all of the (projective) coordinates of all of the torsion points of all abelian varieties defined over $\mathbb{Q}$. Is $L$ algebraically ...

**0**

votes

**0**answers

70 views

### Equidimensionality of stalks of $\operatorname{Proj} S$ when $S$ is equidimensional.

I would like to know a reference of the following statement (or counter example).
Let $S$ be a (commutative) Noetherian standard graded ring over a local ring, i.e., $S = S_0[S_1]$, where $S_0$ is ...

**2**

votes

**1**answer

151 views

### Finite generation of certain $\mathcal{O}_X$-algebra

It is proved in this paper by Kawamata (Theorem 6.1) that for a 3-dimensional normal algebraic variety $X$ which has at most canonical singularities, and a Weil divisor $D$ on it, the ...

**11**

votes

**0**answers

222 views

### Am I missing something about this notion of Mirror Symmetry for abelian varieties?

This is a continuation of my recent question: Mirror symmetry for polarized abelian surfaces and Shioda-Inose K3s.
In the comments of the question, I was directed to the paper ...

**0**

votes

**0**answers

89 views

### “Non-symmetric” Polarized Hodge Structure?

In the definition of a polarized Hodge structure (see here for a definition) is the assumption that the generalized Hodge--Riemann pairing is symmetric up to a possible sign. Does anybody work with ...

**3**

votes

**1**answer

230 views

### Recent ideas in Macaulayfication?

Kawasaki has shown that a quasi-projective variety over a field has a Macaulayfication. His construction does not preserve the Cohen-Macaulay locus of the original variety, only a finite set of ...

**0**

votes

**0**answers

106 views

### Solution to system of polynomial equations

Suppose $A$ is a set of polynomials:$$P_1(x,y_1,\dots,y_n)=0,$$
$$P_2(x,y_1,\dots,y_n)=0,$$
$$\vdots$$
$$P_k(x,y_1,\dots,y_n)=0$$
is a system of equations with coefficients over $\mathbb{Z}$, and ...

**1**

vote

**0**answers

88 views

### A question on the secondary fan

I am studying the secondary fan decomposition of the effective cone of a projective variety $X$. Let as assume that $X$ is a Mori Dream Space. As far as I understand passing from a cone of maximal ...

**0**

votes

**1**answer

131 views

### Cartier divisor on a double cover

Suppose $X$ is a projective variety, and $D$ is a Cartier divisor on $X$. Is it possible (or this is always true) that there is a (ramified) double cover $\pi: \tilde{X} \to X$ such that ...

**2**

votes

**1**answer

376 views

### Symplectic quotient of projective variety is projective?

Let $G$ be a compact connected Lie group and $\mathfrak g^*$ be dual of Lie algebra $\mathfrak g$. Let $M$ be a compact projective variety and $G$ act on $M$ freely and $M$ is $G$ equivariant, and ...

**1**

vote

**0**answers

218 views

### Affine communication lemma and finite limits in the category of rings

Let $X$ be a scheme and $\mathrm{Spec}(B) = V \subseteq X$ be an open affine subset. When using the affine communication lemma (c.f. Theorem 6.3.2, Vakil's notes, Foundations of Algebraic Geometry), ...

**4**

votes

**1**answer

165 views

### Intersections of hypersurfaces of degree $d$ in $\mathbb CP^n$

Is it true that every projective sub-variety of degree $d$ in $\mathbb CP^n$ is an intersection of some number of hypersurfaces of degree $d$? Is there some simple proof of this fact? (I believe this ...

**1**

vote

**0**answers

199 views

### On the Hitchin fibration

I will refer to Simpson's "Higgs bundles and local systems".
Proposition 1.4:
When $X$ is a smooth projective variety, one can build up the moduli space $\mathcal{M}(X,r)$ of rank $r$ Higgs ...

**4**

votes

**0**answers

103 views

### Is this $S$-birational map an open immersion on its domain of definition?

My question is about a claim on the bottom of p. 121 of the book "Neron models" by Bosch, Lutkebohmert, and Raynaud, so I will freely use the general terminology recalled in this book, but will ...

**1**

vote

**1**answer

129 views

### ideal of maximal minors is cohen-macaulay?

Let $k$ be an algebraically closed field.
Let $A$ be an $m \times n$ matrix with linear forms $a_{ij} \in k[x_1, \ldots, x_p]_1$ as entries. Let $I$ be the ideal generated by the maximal minors of ...

**1**

vote

**0**answers

108 views

### Proper monomorphisms in complex analytic spaces

In the algebraic geometry of schemes, we know that monomorphisms which are universally closed (= every pullback is a closed map) and of finite type are closed immersions. See Gortz, Wedhorn "Algebraic ...

**1**

vote

**0**answers

102 views

### How to see that this pairing of line bundles is multiplicative?

Given a projective flat morphism $p: X \rightarrow Y$ of integral noetherian schemes of relative dimension one.
For a coherent sheaf $F$ on $Y$ we can define a line bundle $det(F)$ on $Y$ and for a ...

**29**

votes

**1**answer

1k views

### Algebraic dependency over $\mathbb{F}_{2}$

Let $f_{1},f_{2},\ldots,f_{n}$ be $n$ polynomials in $\mathbb{F}_{2}[x_{1},x_{2},\ldots,x_{n}]$
such that $\forall a=(a_1,a_2,\ldots,a_n)\in\mathbb{F}_{2}^{n}$ we have $\forall ...

**12**

votes

**1**answer

389 views

### A funny factorization of the Jacobian coming from the lines on the Fermat cubic

Here is something which came up in my algebraic geometry class, and I'm wondering if it has a deeper explanation. Let $F(w,x,y,z) = w^3+x^3+y^3+z^3$ and let $X$ be the cubic surface in $\mathbb{P}^3$ ...

**29**

votes

**4**answers

2k views

### Fermat's last theorem over larger fields

Fermat's last theorem implies that the number of solutions of $x^5 + y^5 = 1$ over $\mathbb{Q}$ is finite.
Is the number of solutions of $x^5 + y^5 = 1$ over $\mathbb{Q}^{\text{ab}}$ finite?
Here ...

**0**

votes

**0**answers

142 views

### Rational curves and Serre's construction

Why rational curves, used in Serre's construction of vector bundles, usually corresponds to unstable bundle? I saw this affirmation in Richard Thomas's paper on an obstructed bundle on a CY threefold.
...

**4**

votes

**0**answers

190 views

### Etale Slice Theorem

I found the Luna's Slice Theorem very Technical. It will be helpful if someone illustrates the geometry involved in the theorem. Also why this theorem so useful?
This is Luna's Slice theorem from a ...

**26**

votes

**4**answers

3k views

### Massive cancellations

Let $A=\{a_1,\ldots,a_k\}$ be a fixed, finite set of reals. Let $S_A(n)$ be the set of all reals that are expressible as the sum of at most $2^n$ terms, where each term is a product of at most $n$ ...

**10**

votes

**2**answers

398 views

### Are there known cases of the Mumford–Tate conjecture that do not use Abelian varieties?

(For a formulation of the Mumford–Tate conjecture, see below.)
The question
As far as I know, all non-trivial known cases of the Mumford–Tate conjecture more or less depend on the Mumford–Tate ...

**0**

votes

**1**answer

115 views

### Stability conditions of coherent sheaves on abelian 3-folds

My work for now consists on understanding stability conditions of coherent sheaves on abelian 3-folds. I have the book by D. Huybrechts (the geometry of moduli spaces of sheaves), But I would like to ...

**0**

votes

**1**answer

136 views

### When a proper morphism of schemes is a closed imbedding?

Let $X$ and $Y$ be finitely presented schemes over $\mathbb{C}$. Let $f\colon X\to Y$ be a proper morphism. Let us assume that for any finitely presented scheme $S$ the induced map
$$Mor_{Sch}(S,X)\to ...

**3**

votes

**1**answer

107 views

### On Serre's problem regarding the injectivity of Albert-Algebra cohomological invariants

In these Lecture Notes http://molle.fernuni-hagen.de/~loos/jordan/archive/cohinv/cohinv.pdf from 2006 by Garibaldi on page 21. 7.5 there is the following open problem mentioned:
Is the map
$g_3 ...

**2**

votes

**2**answers

232 views

### Is there an Oka-Grauert principle for homogeneous spaces?

Suppose we have a fibration over the punctured disc (i.e., a deformation of complex manifolds) such that each fiber is a homogeneous space. Is the total space a product of a fiber with the punctured ...

**1**

vote

**1**answer

188 views

### Monodromy of a punctured disc

I meet the following problem which I think related to the monodromy:
Let $D: = \{z \mid |z|<1 \}$ be a disc, and $U \to D$ be a variety fibred over $D$. For each point $t \in D \backslash \{0\}$, ...

**3**

votes

**0**answers

144 views

### Log schemes, differentials, Beilinson

Let $K$ be a $p$-adic field and $K'$ be a finite extension of $K$. Let $\Omega_{(K',\mathcal{O}_{K'})}$ be the sheaf of relative log differentials of the pair $(K',\mathcal{O}_{K'})$ over ...

**1**

vote

**2**answers

185 views

### Computing the nonsingular projective model of a plane curve

Is there an implemented algorithm available in standard software systems (Sage, Magma, Macaulay, etc.) that will compute the nonsingular projective model of a plane curve over $\mathbb Q$?

**0**

votes

**0**answers

141 views

### Rational multiple of a line bundle

In the paper http://arxiv.org/pdf/1207.5011.pdf of Chi Li and Song Sun, they say that "$D$ is a smooth divisor which is $\mathbb{Q}$-linearly equivalent to $−\lambda K_X$ for some $λ \in \mathbb{Q}$", ...

**16**

votes

**0**answers

253 views

### Is there a cotangent bundle of a stable $\infty$-category?

Let $C$ be a stable $\infty$-category. Is there any categorical construction $C \mapsto T^* C$, where $T^* C$ is another stable $\infty$-category, that specializes to the following?
When $C$ is the ...

**0**

votes

**0**answers

130 views

### Regular point of a map in algebraic geometry

What is the correct definition of a regular point of a map in algebraic geometry?
More specifically, let $f:X\to Y$ be a map of varieties with $f(p) = q$, and let $Z=f^{-1}(q)$. Let $\hat{X}$ be the ...

**2**

votes

**0**answers

125 views

### bound on degree of certain polynomials

Consider $m$ polynomials $f_i$ for $i=\{1,\dots,m\}$ in $\Bbb R[x_1,\dots,x_n]$ of degree $d_i$ such that $$Z(f_i)\bigcap Z((p))\neq \emptyset$$ for the sphere $p$ passing through $\{0,1\}^n$. What is ...

**2**

votes

**2**answers

451 views

### Hartshorne Proposition III 8.1

In the proposition mentioned in the title, Hartshorne states that for each $i\geq 0$ the higher direct image sheaf $R^if_*\mathcal F$ is exactly the sheafification of the presheaf given by
$V \mapsto ...

**5**

votes

**1**answer

363 views

### degree of polynomials in nullstellensatz

$(A)$ If $K$ is algebraically closed field, then consider $m$ polynomials $f_i$ for $i=\{1,\dots,m\}$ in $K[x_1,\dots,x_n]$ of degree $d_i$ each with no common root. We know there exists $g_i$ for ...

**1**

vote

**0**answers

75 views

### A formal local triviality statement for smooth maps

Let $f:X\to Y$ be a smooth morphism of schemes of finite type over a field $k$, and suppose that $f(p) = q$. Let $Z = f^{-1}(q)$ be the fiber of $f$. Let $\hat{X}$ be the formal completion of $X$ at ...

**7**

votes

**0**answers

201 views

### Mirror symmetry for polarized abelian surfaces and Shioda-Inose K3s

It is well known (cf. Dolgachev) that there is a beautiful notion of mirror symmetry for lattice-polarized K3 surfaces. That is, if we are given a rank $r$ lattice $M$ of signature $(1, r - 1)$ and a ...

**4**

votes

**1**answer

332 views

### Push-forward of locally free sheaves

Let $X, Y$ be smooth projective varieties and $f:X \times Y \to Y$ be the natural projetion map. Let $\mathcal{F}$ be a locally free sheaf on $X \times Y$. Is it true that $f_*\mathcal{F}$ is locally ...

**0**

votes

**0**answers

104 views

### When can one find holomorphic sections vanishing at a point to a certain order?

Let $X$ be a compact complex manifold (say of dimension $2$) and $L \rightarrow X $ a holomorphic line bundle. Consider the following statements:
Statement $A_0$: Given any point $p\in X$, there ...

**1**

vote

**1**answer

84 views

### Piercing of subspaces in a projective space?

The "piercing subspace" problem may be stated as follows:
There are given several subspaces in a projective space, rather non-intersecting.
Find an additional subspace of a prescribed dimension that ...

**11**

votes

**4**answers

2k views

### Robotics, Cryptography, and Genetics applications of Grothendieck's work? [closed]

I was reading about the passing of Alexander Grothendieck, and something caught my interest:
Mr. Grothendieck was able to answer concrete questions about these relationships by finding universal ...

**1**

vote

**1**answer

188 views

### Smoothness and smoothness over formal neighborhood

Let $f:X\rightarrow Y$ a locally finitely presented map. Let $x\in X$ and $y=f(x)$.
We assume that the map on the level of fomal neighborhoods $X_{x}\rightarrow Y_{y}$ is formally smooth, can we find ...

**6**

votes

**1**answer

209 views

### Do complex tori contain quasi-projective open subsets?

Complex tori are not associated to projective varieties in general.
But can one find an open $U$ inside a complex torus $\mathbb C^g/L$ such that $U$ is the analytification of a quasi-projective ...

**0**

votes

**1**answer

159 views

### A condition on isolated singularity

Suppose $F: {\mathbb C}^N \to {\mathbb C}$ defines a singularity at the origin (for simplicity one can assume that $F$ is a quasi-homogeneous polynomial). Suppose it is nondegenerate, i.e., $dF(z) = ...

**4**

votes

**1**answer

230 views

### Picard of the product of two curves

Can anyone point to me where I can find the proof that the Picard group of the product of two curves is isomorphic to the product of the Picard groups times the hom among the Jacobians?
Does the ...

**0**

votes

**0**answers

78 views

### About stable maps passing through fixed points

In "Notes On Stable Maps and Quantum Cohomology", Fulton and Pandharipande present some results, and their proofs, about the representability of the functor $\mathcal{M}_{g, n}(X, \beta)$, which maps ...

**2**

votes

**0**answers

65 views

### Locus of line bundles with given base points on a curve

Let $C$ be a smooth projective curve over an algebraically closed field. If $D$ is an effective divisor on $C$ (let's say reduced to make things easier) of degree $m$ and $d>m$, is the dimension of ...

**0**

votes

**0**answers

103 views

### An integrality theorem for immersions of quaternion projective spaces in the euclidean space

There are three questions:
Please let me know your proof of the following theorem:
If $\mathbb HP^2$ can be immersed in $\mathbb R^{12}$ with an Euler class $W_{4}(\nu)$ for the normal bundle of ...