949 research outputs found
From SICs and MUBs to Eddington
This is a survey of some very old knowledge about Mutually Unbiased Bases
(MUB) and Symmetric Informationally Complete POVMs (SIC). In prime dimensions
the former are closely tied to an elliptic normal curve symmetric under the
Heisenberg group, while the latter are believed to be orbits under the
Heisenberg group in all dimensions. In dimensions 3 and 4 the SICs are
understandable in terms of elliptic curves, but a general statement escapes us.
The geometry of the SICs in 3 and 4 dimensions is discussed in some detail.Comment: 12 pages; from the Festschrift for Tony Sudber
The modularity of the Barth-Nieto quintic and its relatives
The moduli space of (1,3)-polarized abelian surfaces with full level-2
structure is birational to a double cover of the Barth-Nieto quintic. Barth and
Nieto have shown that these varieties have Calabi-Yau models Z and Y,
respectively. In this paper we apply the Weil conjectures to show that Y and Z
are rigid and we prove that the L-function of their common third \'etale
cohomology group is modular, as predicted by a conjecture of Fontaine and
Mazur. The corresponding modular form is the unique normalized cusp form of
weight 4 for the group \Gamma_1(6). By Tate's conjecture, this should imply
that Y, the fibred square of the universal elliptic curve S_1(6), and Verrill's
rigid Calabi-Yau Z_{A_3}, which all have the same L-function, are in
correspondence over Q. We show that this is indeed the case by giving explicit
maps.Comment: 30 pages, Latex2
Projective varieties with many degenerate subvarieties
We study the problem of classifying the irreducible projective varieties
of dimension in which contain an algebraic family \Cal F
of dimension () of subvarieties of dimension , each one
contained in a . We prove that one of the following happens:
(i) there exists an integer , such that is contained in a
variety of dimension at most containing a family of dimension
of subvarieties of dimension , each one contained in a linear space of
dimension ; (ii) The degree of is bounded by a function of and
(in this case is called of isolated type). Successively we study some
special cases; in particular we give a complete classification of surfaces in
containing a family of dimension of curves of .Comment: 19 pages, AMS-TeX 2.
Tropicalization of classical moduli spaces
The image of the complement of a hyperplane arrangement under a monomial map
can be tropicalized combinatorially using matroid theory. We apply this to
classical moduli spaces that are associated with complex reflection
arrangements. Starting from modular curves, we visit the Segre cubic, the Igusa
quartic, and moduli of marked del Pezzo surfaces of degrees 2 and 3. Our
primary example is the Burkhardt quartic, whose tropicalization is a
3-dimensional fan in 39-dimensional space. This effectuates a synthesis of
concrete and abstract approaches to tropical moduli of genus 2 curves.Comment: 33 page
On complex and real identifiability of tensors
We report about the state of the art on complex and real generic
identifiability of tensors, we describe some of our recent results obtained in
[6] and we present perspectives on the subject.Comment: To appear on Rivista di Matematica dell'Universit\`a di Parma, Volume
8, Number 2, 2017, pages 367-37
Hurwitz numbers and intersections on moduli spaces of curves
This article is an extended version of preprint math.AG/9902104. We find an
explicit formula for the number of topologically different ramified coverings
of a sphere by a genus g surface with only one complicated branching point in
terms of Hodge integrals over the moduli space of genus g curves with marked
points.Comment: 30 pages (AMSTeX). Minor typos are correcte
- …