A well known result in algebraic geometry is that smooth cubic surfaces contain exactly 27 lines over an algebraically closed field. This is no longer true over arbitrary fields. For example, real cubic surfaces contain either 3, 7, 15, or 27 lines. Nevertheless, these lines fall into the class of elliptic lines or hyperbolic lines, and (the number of hyperbolic lines) - (the number of elliptic lines) = 3. In their paper "An arithmetic count of the lines on a smooth cubic surface", Jesse Leo Kass and Kirsten Wickelgren extend this count to smooth cubic surfaces over arbitrary fields. This talk will be a summary of Kass and Wickelgren's results.

An arithmetic count of the lines on a smooth cubic surface

Apr. 27, 2023 4pm (MATH 220)

Init Conditions

Henry Maher

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The Yoneda lemma tells us a great deal about the structure of presheaves, but there is more to be said; I will discuss a pair of theorems that provide even more insight into the structure of presheaf categories.