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Oberseminar SoSe 2011 Zusammenfassung des Vortrags von Remke Kloosterman (HU Berlin)

Mordell-Weil groups, Syzygies and Brill-Noether theory

For a particular class of elliptic threefolds with base P2 we discuss the relation between the rank of the Mordell-Weil group and the syzygies of the singular locus of the discriminant curve of the elliptic fibration. From this we deduce that for each high rank elliptic threefold we find examples of triples (g,k,n) such that the locus

{ [C] ∈ Mg | C admits a g26k and the image of C has at least n cusps}

has much bigger dimension than expected.