MATH Seminar

Title: Non-orientable enumerative problems in $\mathbf{A}^{1}$-homotopy theory
Seminar: Number Theory
Speaker: Andrew Kobin of Emory University
Contact: David Zureick-Brown, dzureic@emory.edu
Date: 2021-11-16 at 4:00PM
Venue: MSC W301
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Abstract: Many enumerative problems in classical algebraic geometry, such as counting lines on a smooth cubic surface, admit a solution over an arbitrary ground field $k$ (of characteristic $\not = 2$) using Morel and Voevodsky's $\mathbf{A}^{1}$-homotopy theory. Recently, several authors have formulated and solved such ``enriched'' enumerative problems using Kass and Wickelgren's ``enriched'' Euler class, which takes values in the Grothendieck--Witt group of $k$ and is only defined when the associated vector bundle is orientable. In joint work with Libby Taylor, we extend Kass--Wickelgren's construction to non-orientable vector bundles using a stacky construction. This allows us to enrich a larger class of enumerative problems, including the count of lines meeting $6$ planes in $\mathbf{P}^{4}$.