Smooth Surfaces with Maximally Many Lines - Janet Page

Title: Smooth Surfaces with Maximally Many Lines

Abstract: How many lines can lie on a smooth surface of degree d? This classical question in algebraic geometry has been studied since at least the mid 1800s, when Clebsch gave an upper bound of d(11d-24) for the number of lines on a smooth surface of degree d over the complex numbers. Since then, Segre and then Bauer and Rams have given sharper upper bounds, the latter of which also holds over fields of characteristic p - d. However, over a field of characteristic p - d, there are smooth projective surfaces of degree d which break these upper bounds. In this talk, I’ll give a new upper bound for the number of lines which can lie on a smooth surface of degree d which holds over any field. In addition, we’ll fully classify those surfaces which attain this upper bound. It turns out that these smooth surfaces are distinguished by their affine cones from the perspective of F-singularities, and we will discuss this connection. This talk is based on joint work with Tim Ryan and Karen Smith and will touch on joint work Anna Brosowsky and with Zhibek Kadyrsizova, Jennifer Kenkel, Jyoti Singh, Adela Vraciu, and Emily Witt.