Hurwitz equivalence of braid monodromies and extremal elliptic surfaces

We discuss the equivalence between the categories of certain ribbon graphs and subgroups of the modular group $\Gamma$ and use it to construct exponentially large families of not Hurwitz equivalent simple braid monodromy factorizations of the same element. As an application, we also obtain exponentially large families of {\it topologically} distinct algebraic objects such as extremal elliptic surfaces, real trigonal curves, and real elliptic surfaces

The generalized Mukai conjecture for symmetric varieties

We associate to any complete spherical variety $X$ a certain nonnegative rational number $\wp(X)$, which we conjecture to satisfy the inequality $\wp(X) \le \operatorname{dim} X - \operatorname{rank} X$ with equality holding if and only if $X$ is isomorphic to a toric variety. We show that, for spherical varieties, our conjecture implies the generalized Mukai conjecture on the pseudo-index of smooth Fano varieties due to Bonavero, Casagrande, Debarre, and Druel. We also deduce from our conjecture a smoothness criterion for spherical varieties. It follows from the work of Pasquier that our conjecture holds for horospherical varieties. We are able to prove our conjecture for symmetric varieties.Comment: 33 pages, 2 figures, 6 table

Dihedral coverings of trigonal curves

We classify and study trigonal curves in Hirzebruch surfaces admitting dihedral Galois coverings. As a consequence, we obtain certain restrictions on the fundamental group of a plane curve~$D$ with a singular point of multiplicity $(\deg D-3)$

Failure of the Regge approach in two dimensional quantum gravity

Regge's method for regularizing euclidean quantum gravity is applied to two dimensional gravity. We use two different strategies to simulate the Regge path integral at a fixed value of the total area: A standard Metropolis simulation combined with a histogramming method and a direct simulation using a Hybrid Monte Carlo algorithm. Using topologies with genus zero and two and a scale invariant integration measure, we show that the Regge method does not reproduce the value of the string susceptibility of the continuum model. We show that the string susceptibility depends strongly on the choice of the measure in the path integral. We argue that the failure of the Regge approach is due to spurious contributions of reparametrization degrees of freedom to the path integral.Comment: 27 pages, LaTex + uuencoded figure files (13 postscript files