Secants to the Kummer Variety

Abstract

In this thesis we mainly prove two results in an algebro-geometric way: If one has a curve Γ of (honest) quadrisecant planes to the Kummer variety of an indecomposable principally abelian variety (X,Θ) then the curve Γ is twice the minimal class, under certain technical geometric conditions. By previous analytic results (see [20]), this will imply that X is a Prym variety. As a generalization of this results, adding one geometric condition we get that having a curve of (m+2)-secants (for a minimal m) implies that the abelian variety has a curve that is m-times the minimal cohomological class. The second result of this thesis is a an answer to a natural generalization of a question Welters asked about trisecants (see [35]) and is as follows: Under certain geometric conditions, does the existence of m different (m+2)-secant m-planes imply that one has a curve of honest (m+2)-secant (m-)planes? We show that under certain conditions, this question has a positive answer (see Theorem 4.4.4).