Examples 1. A K3 surface of degree two is a double cover of P, ramified in a smooth sextic. K3 surfaces of degree four are smooth quartics in P. A K3 surface of degree six is a smooth complete intersection of a quadric and a cubic in P. And, finally, K3 surfaces of degree eight are smooth complete intersections of three quadrics in P. The Picard group of a K3 surface is isomorphic to Zn where n...

Preface Why we called the class of two-dimensional Shimura varieties, which are not Hilbert modular, "Picard modular surfaces" ? In the mean time the name has been generally accepted, see e.g. Langlands (and others) L-R]. On the one hand Picard worked on special Fuchsian systems of diierential equations; on the other hand Shimura Shi] introduced and investigated moduli spaces of abelian varieti...

A differential Azumaya algebra, and in particular a differential matrix algebra, over a differential field K with constants C is trivialized by a Picard–Vessiot (differential Galois) extension E. This yields a bijection between isomorphism classes of differential algebras and Picard–Vessiot cocycles Z(G(E/K), PGLn(C)) which cobound in Z (G(E/K), PGLn(E)).

Using Moriwaki’s calculation of the Q-Picard group for the moduli space of curves, I prove the strong Franchetta Conjecture in all characteristics. That is, the canonical class generates the group of rational points on the Picard scheme for the generic curve of genus g ≥ 3. Similar results hold for generic pointed curves.

1 My thanks to Serge Pahaut for our many fruitful discussions together over the past few years. Nathalie Picard and Jean Picard submitted useful suggestions for improving the quality and readability of this paper. Lastly, Kurt van Dender also helped me to improve its presentation.

We construct full strong exceptional collections of line bundles on smooth toric Fano Deligne-Mumford stacks of Picard number at most two and of any Picard number in dimension two. It is hoped that the approach of this paper will eventually lead to the proof of the existence of such collections on all smooth toric nef-Fano Deligne-Mumford stacks.

We define and study the 2-category of torsors over a Picard groupoid, a central extension of a group by a Picard groupoid, and commutator maps in this central extension. Using this in the context of two-dimensional local fields and two-dimensional adèle theory we obtain the two-dimensional tame symbol and a new proof of Parshin reciprocity laws on an algebraic surface.

We determine the number of cusps of minimal Picard modular surfaces. The proof also counts cusps of other Picard modular surfaces of arithmetic interest. Consequently, for each N > 0 there are finitely many commensurability classes of nonuniform arithmetic lattices in SU(2, 1) that contain an N -cusped surface. We also discuss a higher-rank analogue.

In this article we present robust, efficient and accurate fully implicit time-stepping schemes and nonlinear solvers for systems of reaction-diffusion equations. The applications of reaction-diffusion systems is abundant in the literature, from modelling pattern formation in developmental biology to cancer research, wound healing, tissue and bone regeneration and cell motility. Therefore, it is...