MIT Number Theory Seminar
Tuesday, September 28, 2021 at 4:30am to 5:30am
Title: Quadratic points on intersections of quadrics
Abstract: A protective degree 𝒹 variety always has a point defined over a degree 𝒹 field extension. For many degree 𝒹 varieties, this is the best possible statement, that is, there exist classes of degree 𝒹 varieties that never have points over extensions of degree less than 𝒹 (nor even over extensions whose degree is nonzero modulo 𝒹). However, there are some classes of degree 𝒹 varieties that obtain points over extensions of smaller degree, for example, degree 9 surfaces in ℙ 9 , and 6-dimensional intersections of quadrics over local fields. In this talk, we explore this question for intersections of quadrics. In particular, we prove that a smooth complete intersection of two quadrics of dimension at least 2 over a number field has index dividing 2, i.e., that it possesses a rational 0-cycle of degree 2. This is joint work with Brendan Creutz.
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