Harkaran Uppal

Lines on del Pezzo surfaces

One of the crowning achievements of classical algebraic geometry is the Cayley–Salmon theorem, which states that any smooth cubic surface over an algebraically closed field contains exactly 27 lines. Over more general fields, however, the situation becomes more subtle: the number of lines depends on the arithmetic of the field. Segre classified the possible numbers of lines that can appear on a cubic surface over arbitrary fields and showed that all such line counts can be realised over the rational numbers.
In this talk, I will discuss joint work with Enis Kaya, Stephen McKean, and Sam Streeter, in which we extend this perspective to del Pezzo surfaces -- a class of surfaces that includes cubic surfaces. We investigate which line counts can occur on del Pezzo surfaces over general fields and how these counts are influenced by the arithmetic of the field. We also explore the analogous question for conic bundles.