Nodal rational sextics in the real projective plane

Abstract

This thesis studies nodal sextics (algebraic curves of degree six), and in particular rational sextics, in the real projective plane. Two such sextics with k nodes are called rigidly isotopic if they can be joined by a path in the space of real nodal sextics with k nodes. The main result of the first part of the thesis is a rigid isotopy classification of real nodal sextics without real nodes, generalizing Nikulin’s classification of non-singular sextics. In the second part we study sextics with real nodes and we describe the rigid isotopy classes of such sextics in the case where the sextics are dividing, i.e., their real part separates the complexification (the set of complex points) into two halves. As a main application, we give a rigid isotopy classification for those nodal real rational sextics which can be perturbed to maximal or next-to-maximal sextics in the sense of Harnack’s inequality. Our approach is based on the study of periods of K3 surfaces, drawing on the Global Torelli Theorem by Piatetski-Shapiro and Shafarevich and Kulikov’s surjectivity theorem, as well as Nikulin’s results on symmetric integral bilinear forms.