We prove a geometric logarithmic derivative lemma for rigid analytic mappings to algebraic varieties in characteristic zero. We use the lemma to give a new and simpler proof (at least in characteristic zero) of Berkovich’s little Picard theorem [Ber, Theorem 4.5.1], which says there are no nonconstant rigid analytic maps from the affine line to non-singular projective curves of positive genus, ...

We are concerned with the numerical exact controllability of semilinear wave equation on interval (0, 1). introduce a Picard iterative scheme yielding sequence approximated solutions which converges towards solution null problem, provided that initial data small enough. The boundary control, is applied at endpoint $$x=1$$ , taken in space $$H^1_0(0,T)$$ for $$T=2$$ . For linear part, control in...

Picard Lattices of Families of K3 Surfaces bysarah-marie belcastro Chair: Igor Dolgachev It is a nontrivial problem to determine the Picard Lattice of a given surface; theobject of this thesis is to compute the Picard Lattices of M. Reid’s list of 95 fami-lies of Gorenstein K3 surfaces which occur as hypersurfaces in weighted projectivespace. Reid’s list arises in many problems;...

We prove a geometric logarithmic derivative lemma for rigid analytic mappings to algebraic varieties in characteristic zero. We use the lemma to give a new and simpler proof (at least in characteristic zero) of Berkovich’s little Picard theorem, which says there are no nonconstant rigid analytic maps from the affine line to nonsingular projective curves of positive genus, and of Cherry’s result...

We study the surface S̄ parametrizing cuboids: it is defined by the equations relating the sides, face diagonals and long diagonal of a rectangular box. It is an open problem whether a ‘rational box’ exists, i.e., a rectangular box all of whose sides, face diagonals and long diagonal have (positive) rational length. The question is equivalent to the existence of nontrivial rational points on S̄. ...

Let X be a K3 surface which is intersection of three (i.e. a net P) of quadrics in P. The curve of degenerate quadrics has degree 6 and defines a natural double covering of P ramified in this curve which is again a K3. This is a classical example of a correspondence between K3 surfaces which is related with moduli of sheaves on K3’s studied by Mukai. When general (for fixed Picard lattices) X a...

A long-standing question in the theory of rational points of algebraic surfaces is whether a K3 surface X over a number field K acquires a Zariski-dense set of L-rational points over some finite extension L/K. In this case, we say X has potential density of rational points. In case XC has Picard rank greater than 1, Bogomolov and Tschinkel [2] have shown in many cases that X has potential densi...