Lines in algebraic geometry


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February 16, 2022 4:30 pm - 5:30 pm
Science Center 507
Address: 1 Oxford Street, Cambridge, MA 02138 USA

Hannah Larson - Stanford

Suppose you write down a general polynomial in x, y, z and consider the surface of all points where it vanishes. What can you say about the family of lines contained in this surface? Are there no lines, a finite number of lines, infinitely many? We'll derive an expected dimension for the family of lines depending on the degree of the polynomial (and generalize this to more variables). In the case of cubic surfaces, we'll discuss some more subtle questions regarding the geometry of lines over the real numbers. This story motivates some results, joint with Isabel Vogt, about a closely related problem concerning bitangents (lines that are tangent twice) to a plane quartic. There will be many examples and "hands on demos."