Schur quadrics, cubic surfaces and rank 2 vector bundles over the projective plane

A cubic surface in $P^3$ is known to contain 27 lines, out of which one can form 36 Schlafli double - sixes i.e., collections $l_1,...,l_6, l'_1,..., l'_6\}$ of 12 lines such that each $l_i$ meets only $l'_j, j\neq i$ and does not meet $l_j, j\neq i$. In 1881 F. Schur proved that any double - six gives rise to a certain quadric $Q$ , called Schur quadric which is characterized as follows: for any $i$ the lines $l_i$ and $l'_i$ are orthogonal with respect to (the quadratic form defining) $Q$. The aim of the paper is to relate Schur's construction to the theory of vector bundles on $P^2$ and to generalize this construction along the lines of the said theory.Comment: 27 pages, plain TE

On the rationality of the moduli space of L\"uroth quartics

We prove that the moduli space M_L of L"uroth quartics in P^2, i.e. the space of quartics which can be circumscribed around a complete pentagon of lines modulo the action of PGL_3(CC) is rational, as is the related moduli space of Bateman seven-tuples of points in P^2.Comment: 7 page

Algebraic entropy and the space of initial values for discrete dynamical systems

A method to calculate the algebraic entropy of a mapping which can be lifted to an isomorphism of a suitable rational surfaces (the space of initial values) are presented. It is shown that the degree of the $n$th iterate of such a mapping is given by its action on the Picard group of the space of initial values. It is also shown that the degree of the $n$th iterate of every Painlev\'e equation in sakai's list is at most $O(n^2)$ and therefore its algebraic entropy is zero.Comment: 10 pages, pLatex fil

Hori--Vafa mirror models for complete intersections in weighted projective spaces and weak Landau--Ginzburg models

We prove that Hori--Vafa mirror models for smooth Fano complete intersections in weighted projective spaces admit an interpretation as Laurent polynomials.Comment: 5 pages; several minor changes has been mad