0
votes
0answers
32 views

Albanese map on a universal jacobian associated to a curve on an abelian surface

Given an abelian surface $A$ and a curve $C$ on it, consider the component $\{C\}$ of the Hilbert scheme of curves with the same cohomology class of $C$. I would say that it has a fibration structure ...
-4
votes
0answers
33 views

What do rank-2 tensor entries represent? What's their geometrical meaning? [on hold]

given a vector (with, say, 3 dimensions), if I multiply that with a second-rank tensor, I can make it change in both direction and magnitude (for that 2nd rank tensors are involved in vector-vector ...
0
votes
0answers
74 views

Cohomology group vs sheaf of cohomology group

Suppose $F$ is a coherent sheaf on a smooth (algebraic or complex) variety $X$. Then we can consider the cohomology groups $$H^p(X,F)$$ for all $i$. Now, let we consider the sheaf ...
1
vote
0answers
43 views

Singularities of secant varieties of rational normal curves

Let $C\subset\mathbb{P}^n$ be a rational normal curve of degree $n$, and let $Sec_k(C)\subset\mathbb{P}^n$ be its $k$-th secant variety. By Theorem 1.1 in this paper: ...
0
votes
1answer
48 views

Is there some lattice not rigid

I heard that in complex hyperbolic space setting for example CH2, there is some deformation of lattice nontrivial. What confused me is it seems contradicting Mostow Rigidity. Could someone explain ...
2
votes
0answers
40 views

Which valuations of a field yield codimension $1$ subschemes of their 'models'

For a field $F$ (for example, a one generated by a finite number of its elements) there is a directed set of its 'models' (in this case those are 'arithmetic' schemes whose fraction field is $F$). It ...
1
vote
0answers
43 views

If a compact real submanifold of $\mathbb{CP}^n$ is approximable by complex algebraic curves, is it algebraic?

To make this into a separate question: If the supports of a sequence of complex algebraic curves in $\mathbb{CP}^n$ (images of non-constant holomorphic maps from compact Riemann surfaces) converge to ...
1
vote
0answers
87 views

Complex structures on Riemann surfaces

This is cross posted from math.SE: http://math.stackexchange.com/q/876432/9 Let $M$ be a Riemann surface and $[\alpha] \in H^{0,1}(M; T^{1,0} M) \simeq (H^0(M;K^2))^*$. Considering $\alpha$ as a map ...
13
votes
0answers
276 views

The lonely molecule

Suppose $n$ air molecules (infinitesimal points) are bouncing around in a unit $d$-dimensional cube, with perfectly elastic wall collisions. Let $k=n^{\frac{1}{d}}$. For example, in 3D, $d=3$, with ...
2
votes
1answer
134 views

When are isotrivial families split by a finite base-change?

A well-known theorem of Grauert and Fischer states that a smooth proper family of complex manifolds is a locally trivial fibration as soon as all the fibers are isomorphic. It is also easy to obtain ...
1
vote
1answer
180 views

Example of a specific manifold

I want to find a example of a manifold that has positive scalar curvature but is not half conformally flat. Does there exists such manifolds? Thanks.
1
vote
0answers
62 views

Question about the specialization map for Etale Fundamental Groups

Let $A$ be a complete, discrete valuation ring, and let $s$ (resp $\eta$) be the special (resp. generic) point of $S=Spec(A)$. Let $\phi:X \rightarrow S$ be a proper morphism and fix geometric base ...
2
votes
1answer
450 views

is cross product of 2 or 3 vectors in 4 D is possible ? why ? i'm an engineer and please kelp me in explaining in a way i can understand? [on hold]

Is cross product of 2 or 3 vectors in 4 D is possible ? why ? i'm an engineer and please kelp me in explaining in a way i can understand ? It is also not clear that why vectors depend on the dimension ...
1
vote
0answers
55 views

Obstruction to Gorenstein Liaisons of space curves

Let $P_1,P$ be Hilbert polynomials of curves in $\mathbb{P}^3$. Denote by $H_{P_1,P}$ the flag Hilbert scheme parametrizing pairs $(C_1 \subset C)$ where $C_1, C$ are of Hilbert polynomials $P_1$ and ...
1
vote
0answers
218 views

Grothendieck's letter to serre

Is it the letter dated in 27/08/1965 of Grothendieck where he presents to Serre the Standard conjectures on algebraic cycles?
1
vote
0answers
63 views

A topological criterion for connectedness of a semi-ample divisor

I have a half page long proof of the following statement, and I would like to know if this is (a corollary of) a well known statement. Maybe there is a reference or a three lines proof? Statement. ...
1
vote
0answers
30 views

Parametrized cancelations in stable Morse theory

Let $B$ be a closed manifold. Let $\pi : M\to B$ be a submersion such that each fiber is a manifold without boundary. Let $f : M \to \mathbb{R}$ be a function such that the restrictions $f_x$ to each ...
0
votes
0answers
37 views

Embedding dimension: local finiteness & intuition for more general spaces

Can every complex analytic space be covered by Stein spaces of finite embedding dimension? I am almost sure that ought to be true. Here the definition of embedding dimension I have in mind is $$ ...
1
vote
1answer
129 views

Examples of nontrivial local systems in Decomposition Theorem

There is a proper map $f: X \rightarrow Y$ of projective varieties. The Decomposition Theorem of Beilinson–Bernstein–Deligne-Gabber states that $$Rf∗IC_X \cong \oplus_a ...
2
votes
3answers
419 views

Reference request: SGA7

I want to read SGA7. Without considering the others SGA and EGA, Which are the textbooks for monodromy theory?
2
votes
0answers
66 views

Update on list of open problems for Cherednik/Symplectic Reflection Algebras

Background: There are two lists of open problems about Cherednik or Symplectic Reflection Algebras from 2007: Ian Gordon's Problems, Chapter 9 in Symplectic Reflection Algebras, and Ginzburg & ...
7
votes
0answers
85 views

Newton polygons of modular polynomials

This is pretty much straightforward curiosity. Is there anything known about Newton polygons of classical modular polynomials (polynomial relations between $j(\tau)$ and $j(n\tau)$)? I understand that ...
5
votes
1answer
184 views

Is the Tate conjecture known for etale covers of products of curves

Let $X$ be a (smooth projective geometrically connected) surface over a finitely generated field $k$. The Tate conjecture predicts that, for $l$ a prime number invertible in $k$, the Chern class map ...
12
votes
2answers
452 views

History of the connection between Riemann surfaces and complex algebraic curves

As noted in the question "Links between Riemann surfaces and algebraic geometry", there are strong connections between Riemann surfaces and algebraic geometry - for example, compact Riemann surfaces ...
2
votes
0answers
43 views

Cusp points in Alexandrov spaces

Given a space of bounded integral curvature (by which I mean a topological surface with an intrinsic metric, such that the sum of excesses of any finite collection of non-overlapping simple triangles ...
2
votes
0answers
31 views

Looking for N-dimensional spheres in the configuration space of the colorful Tverberg problem

Here we use standard notation for Tverberg's theorem: Dimension $d$, number of partition blocks $r$, and $N=(r-1)(d+1)$. The configuration space of Tverberg's theorem is the simplicial complex ...
9
votes
1answer
166 views

Borel-Serre compactification of $\mathbb{H}^3 / SL_2(\mathcal{O}_K)$

Let $\mathbb{H}^3$ be the three-dimensional hyperbolic space. Let $K$ be an imaginary quadratic number field and $\mathcal{O}_K$ its ring of integers. Then $SL_2(\mathcal{O}_K)$ acts on $\mathbb{H}^3$ ...
2
votes
0answers
29 views

Generators for lattice polytopes

Let $V$ be a finite set of integral vectors in $\mathbb{R}_n$. Consider the real (positive) convex cone $\mathbb{R}^+V$ spanned by $V$. We can assume that all vectors in $V$ are extremal in ...
2
votes
0answers
121 views

Universal cover of elliptic curve without a point

This is something that I perhaps would be expected to know, but don't. Let $E_\tau$ be the elliptic curve ${\mathbb C}/({\mathbb Z +\mathbb Z} \tau)$. Consider the complement of a point in $E_\tau$, ...
2
votes
0answers
64 views

Degree of join of two varieties‏

Let $X\subset\mathbb{P}^n$ be an irreducible variety of degree $d$ and dimension $n-4$. Let $L\subset X$ be a line of mulitplicity $m$ in $X$. Assume that $m+1\leq d$ so that the general line spanned ...
0
votes
0answers
29 views

Any software that can symmetrize input sets?

Is there any software that contains symmetrization techniques ex. polarization, Steiner Symmetrization etc. I suppose not. Which software would you suggest for rigid transformations? Thank you
14
votes
0answers
271 views

function field analogy and global/absolute geometry

The "function field analogy" seems to be a topic that is considerably bigger than any one existing writeup conveys. There are several old question on MO and and MathSE that ask for details. One of the ...
4
votes
0answers
64 views

Symplectic sum and Symplectic cut

The symplectic sum of Gompf and the symplectic cut of Lerman are known to be inverse of each other, in the sense that if you apply one of these first and the other one afterward, you obtain the ...
0
votes
1answer
204 views

Computing the Chern class of $S^6$ [on hold]

I am trying to calculate the Chern class of the tangent bundle of the sphere $S^6$. I am told that this is an interesting case, since $S^6$ is not a complex manifold, but it has an almost complex ...
0
votes
0answers
65 views

Examples of manifolds whose second Stiefel-Whitney satisfies a nontriviality condition

I'm looking for examples of pairs $(M,L)$ where $M$ is a symplectic manifold, $L$ a (closed, connected) Lagrangian submanifold, such that the second Stiefel-Whitney of $L$, $w_2(TL)$, evaluates ...
3
votes
2answers
177 views

Is the hypersurface satisfying $\langle x-x_0,\nu\rangle>0$ diffeomorphic to sphere?

Let $p:M\to \mathbb{R}^{n+1}$ be the closed immersed hypersurface. Is the following thing right? If there exists a point $x_0$ in $\mathbb{R}^{n+1}$ such that $\langle x(p)-x_0,\nu(p)\rangle>0$ ...
3
votes
0answers
82 views

Lagrangian submanifolds in $T^\ast S^n$

Let $L\subset T^\ast S^n$ be a properly embedded Lagrangian submanifold homeomorphic to $\mathbb{R}^n$ with respect to the canonical symplectic structure on $T^\ast S^n$, and suppose $L$ intersects ...
2
votes
0answers
72 views

Symplectic isotopies between small balls?

Let $(X,\omega)$ be a connected symplectic manifold, possibly with boundary. Let $g_1, g_2: B(1) \to X$ be two balls in $X$. Is it true that if $\delta$ is sufficiently small, then there is an ...
7
votes
3answers
515 views

nth term in the Baker-Campbell-Hausdorff formula

I am trying to prove a result for which I need the nth term of the Baker-Campbell-Hausdorff formula. I came at this particular result (which is not of significance for the question, but mentioning for ...
1
vote
0answers
81 views

Interpretation of the Gross-Zagier formula for Green function

I am reading the paper of Gross and Zagier on heights of Heegner points and would like to check with the experts whether the following (meta?)mathematical statement makes sense. In the calculation of ...
4
votes
1answer
193 views

Noncommutative HKR theorem

What is the analog of HKR theorem in the noncommutative world? Recall that the well-known theorem by Hochschild-Kostant-Rosenberg says that for a smooth commutative algebra $A$ of finite type ...
-2
votes
2answers
79 views

Hexagon Formed by connecting Trisections of triangle sides [on hold]

Is there a theorem for the area of the hexagon formed by connecting the points formed when the sides of a triangle are trisected? It appears that the ratio of the area of the triangle to the area of ...
2
votes
2answers
164 views

Example of linearization for GIT

Take a vector space $V$ (finite dimensional, over the complex numbers), let $G=SL(V)$. The group $G$ acts on $\mathbb{P}V$ and we can linearize its action to an action on the line bundle ...
4
votes
1answer
222 views

Non-vanishing of elements in cohomology of full Flag varieties

Consider the full flag variety $F_n$ consisting of full flags in $\mathbb C^n$. There is a collection of tautological bundles on $F_n$: $0=U_0\subset U_1\subset ...\subset U_{n-1}\subset U_n=\mathbb ...
2
votes
2answers
204 views

Basic questions on the Hilbert scheme/ Douady space

Let $X$ be a complex projective scheme (resp. complex analytic space). The Hilbert scheme (resp. Douady space) parameterizes closed subschemes (resp. complex analytic subspaces) of $X$. More ...
9
votes
3answers
1k views

Weil's Riemann Hypothesis for dummies?

Weil's Riemann Hypothesis is a deep result that I don't fully understand, but it has understandable corollaries which interest me. For example: (a) For any projective curve $X$ satisfying certain ...
2
votes
2answers
256 views

Canonical Sheaf of Projective Space

I am stuck on one step that occurs without explanation in several Algebraic geometry books. Starting from the exact sequence $$0\rightarrow \Omega_{\mathbb{P}^n}\rightarrow ...
12
votes
1answer
547 views

Is there any progress on Problem 12 (from Schoen and Yau)?

I saw the following question from the "Problem Section" in Schoen and Yau, page 281, problem 12: Let $M_1, M_2$ each have negative curvature. If $\pi_1 (M_1)=\pi_1 (M_2)$, prove that $M_1$ is ...
1
vote
2answers
198 views

commutative algebra, diagonal morphism

can anyone help me with the following statement (it is part of a bigger proof where it is not explained). Let $B$ be a finite type $A$-algebra and consider the kernel $I$ of the diagonal ...
2
votes
0answers
49 views

calculating the Hilbert polynomial of a scheme given its primary decomposition

Given a scheme $X$ in $\mathbb{P}^n$, let $I_X$ be its, saturated, associated ideal. Suppose that a primary decomposition of this ideal is given by $$ I_X =I_1 \cap \ldots \cap I_2 $$ I was ...