4
votes
2answers
383 views

Do constructible sets have Krull dimension?

Let $(I,\leq)$ be a poset. Recall that the Krull dimension of $I$ is defined as follows: -- $K.dim(I)=-1$ if and only if $I=\{0\}$; -- if $\alpha$ is an ordinal and we already defined what it means ...
2
votes
1answer
270 views

Automorphism groups of indefinite non-unimodular integer lattices

Does anyone know of any papers in which structural aspects of the orthogonal group of some indefinite non-unimodular integral lattice are calculated? The exact lattice isn't so important and they ...
1
vote
1answer
110 views

Lattice basis with Gram-Schmidt vectors of increasing length

Let $\Lambda$ be an $n$-dimensional lattice in $\mathbb R^n$ and let $\cal B$ be the set of all bases that generate $\Lambda$. For a basis $\mathbf{B}=[\mathbf{b}_1, ... ,\mathbf{b}_n]\in {\cal B}$, ...
4
votes
1answer
278 views

Invariant lattice of algebraic surface.

Given an algebraic surface $S$ with action of a finite group $G$. Is it true that the invariant lattice $H^2(X,\mathbb{Z})^G$ is generated by elements pulled back from the $H^2(X/G,\mathbb{Z})$ (or ...
0
votes
0answers
139 views

$T^2$-fibered K3 surface with involution

Let $S$ be a K3 surface and $f:S\rightarrow \mathbb{P}^1$ a $T^2$-fibration (not necessarily holomorphic, I have a special Langrangian fibration in mind). Assume there is a $k$-section, then a fiber ...
2
votes
0answers
238 views

Singular fibers of an elliptic fibered K3 surface.

Let $f:S\rightarrow \mathbb{P}^1$ be an elliptic K3 surface. Assume that $\mathrm{Pic}(S)\cong U$, where $U$ stands for the hyperbolic lattice. I think that the elliptic fibration has only singular ...
3
votes
0answers
83 views

Versions of Helly's Theorem for Unbounded Parallelpipeds

I'm studying Frobenius splitting of toric varieties based on Sam Payne's characterization of Frobenius splitting ("Frobenius splittings of toric varieties" (2008)) and I came across a sort of Helly's ...
0
votes
1answer
505 views

Neron-Severi Lattice of Elliptic K3

I'm trying to compute Neron-Severi lattices of some K3 surfaces. They have elliptic fibrations with multiple sections. Setting one section to be the identity section, I can write down a Weierstrass ...
11
votes
3answers
1k views

When are Ehrhart functions of compact convex sets polynomials?

Given a lattice $L$ and a subset $P\subset \mathbb R^d$, we define for each positive integer $t$ $$f_P(L,t)=|(tP\cap L)|$$ the number of lattice points in $tP$. Let's say $P$ is nice if $f_P(L,t)$ is ...
2
votes
1answer
566 views

Theta Functions and Cousins

So I am (barely) familiar with the construction of the theta function of an integral lattice L. The theta function, as I understand it, is defined as the function which takes a variable z and spits ...