International Journal of Mathematics

Self-rational maps of generic algebraic K3 surfaces are conjectured to be trivial. We relate this conjecture to a conjecture concerning the irreducibility of the universal Severi varieties parameterizing nodal curves of given genus and degree lying on some K3 surface. We also establish a number of numerical constraints satisfied by such nontrivial rational maps, that is of topological degree > 1.

We use open book representations of contact 3-manifolds to compute the cylindrical contact homology of a Stein-fillable contact 3-manifold represented by the open book whose monodromy is a positive Dehn twist on a torus with boundary.

We describe open book decompositions of links of simple surface singularities that support the corresponding unique Milnor fillable contact structures. The open books we describe are isotopic to Milnor open books.

We prove that the gonality among the smooth curves in a complete linear system on a K3 surface is constant except for the Donagi–Morrison example. This was proved by Ciliberto and Pareschi under the additional condition that the linear system is ample. The constancy was originally conjectured by Harris and Mumford.

As a consequence we prove that exceptional curves on K3 surfaces satisfy the Eisenbud–Lange–Martens–Schreyer conjecture and explicitly describe such curves. They turn out to be natural extensions of the Eisenbud–Lange–Martens–Schreyer examples of exceptional curves on K3 surfaces.

An analytic pair of dimension n and center V is a pair (V, M) where M is a complex manifold of (complex) dimension n and V ⊂ M is a closed totally real analytic submanifold of dimension n. To an analytic pair (V, M) we associate the class of the functions u : M → [0, π/4] which are plurisubharmonic in M and such that u(p) = 0 for each p ∈ V. If admits a maximal function u, the triple (V, M, u) is said to be a maximal plurisubharmonic model. After defining a pseudo-metric E_{V, M} on the center V of an analytic pair (V, M) we prove (see Theorem 4.1, Theorem 5.1) that maximal plurisubharmonic models provide a natural generalization of the Monge–Ampère models introduced by Lempert and Szöke in [18].