Maximal linear spaces contained in the base loci of pencils of quadrics

By Xiaoheng Wang. 2013

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The geometry of the Fano scheme of maximal linear spaces contained in the base locus of a pencil of quadrics has been studied by algebraic geometers when the base field is algebraically closed. In this paper, we work over an arbitrary base field of characteristic not equal to 2 and show how these Fano schemes are related to the Jacobians of hyperelliptic curves. In particular, if $B$ is the base locus of a generic pencil of quadrics in $\bbp^{2n+1}$, and $F$ is the Fano variety of $n - 1$ planes contained in $B$, then $F$ is a component of a disconnected commutative algebraic group $G = \picz(C) \dcup F \dcup \pico(C) \dcup F'$, where $C$ is the hyperelliptic curve defined by the discriminant form of the pencil. In the second half of this paper, we study regular pencils of quadrics, where the hyperelliptic curve defined by the discriminant is singular.

Subfields: Computational Theory and Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/f4c18961c58021eac423659d7b93860907c1f1bd

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The average size of the 2-Selmer group of Jacobians of hyperelliptic curves having a rational Weierstrass point (2012, background): “deed, one can deﬁne the structure of a commutative algebraic group on t he disconnected variety G = J∪FT such that the involution −1G of G induces the involution τ on the non-trivial componentFT (see [51]). We prove that the isomorphism class of the principal homogeneous space PT over Qdetermines th”