Seshadri constants and geometry of surfaces

In this work we study Seshadri constants on ruled surfaces. In the case of P1 × P1 we study Riemann-Roch expected curves in the context of the Nagata-Biran Conjecture. This conjecture predicts that for a sufficiently large number of points multiple point Seshadri constants of an ample line bundle on algebraic surface are maximal. Biran gives an effective lover bound N0. We construct examples verifying to the effect that the assertions of the Nagata-Biran Conjecture can not hold for small number of points. We observe also that there is a strong connection between the Riemann-Roch expected curves on P1 × P1 and the symplectic packing problem. Biran relates the packing problem to the existence of solutions of certain Diophantine equations. We construct such solutions for any ample line bundle on P1 × P1 and a relatively small number of points. These solutions geometrically correspond to Riemann-Roch expected curves. Finally we discuss in how far the Biran number N0 is optimal in the case P1×P1. In fact we conjecture that it can be replaced by a lower number and we provide evidence justifying this conjecture. Coming from the other end, motivated by Hwang-Keum, Szemberg-Tutaj-Gasi´nska we study impact of (low) Seshadri constants on the geometry of the underlying surface. First we study multiple point Seshadri constant and give a sharp upper bound on Seshadri constants resulting in detecting a fiber structure of the surface. Such a bound was given in the case of single point Seshadri constants by Szemberg and Tutaj-Gasi´nska. In that case we show that the only example satisfying their bound is a cubic surface in P3 and thus a better bound holds for all other surfaces.



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