Geometry of Extremal Surfaces

Abstract: Mathematicians have long been fascinated by the lines contained on surfaces. For example, surfaces of degree one and two contain infinitely many lines, but in degree three, a smooth surface over an algebraically closed field famously contains 27 lines. In degrees four and higher, a general smooth surface has no lines, and Segre showed that over the complex numbers a smooth surface of degree $d$ contains at most $(d-2)(11d-6)$ lines. In contrast, in positive characteristic, there exist smooth surfaces which wildly violate Segre’s bound. These surfaces fit into a class of varieties which were recently defined and studied for some of their other "extremal" properties using commutative algebraic tools. In this talk, I will discuss some recent results on the geometry of "extremal" surfaces, including on the combinatorics of their lines in the smooth case, which, besides being abundant, also form fascinating configurations. Time permitting, I will also discuss the geometry of singular extremal surfaces.

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