Curves and symmetric spaces. I. (English) Zbl 0871.14025

Mukai, Shigeru

Am. J. Math. 117, No. 6, 1627-1644 (1995).

This is the first part of a series of two papers in which the author presents detailed proofs of the results previously announced [S. Mukai, Proc. Japan Acad., Ser. A 68, 7-10 (1992; Zbl 0768.14014)]. In this part one proves the following main result:
Theorem. A curve \(C\) of genus 7 is a transversal linear section of the 10-dimensional orthogonal Grassmannian \(X\subset \mathbb{P}^{15}\) if and only if \(C\) has no \(g^1_4\). Moreover, the transversal linear subspaces which cut out \(C\) are unique up to the action of \(SO(10)\).
As a corollary the author gets the following: If a curve \(C\) of genus 7 is cut out from \(X \subset \mathbb{P}^{15}\) by a transversal linear subspace \(P\) then every automorphism of \(C\) is the restriction of an automorphism of \(X\subset \mathbb{P}^{15}\) which preserves \(P\).

Reviewer: L.Bădescu (Bucureşti)

MSC:

14H45	Special algebraic curves and curves of low genus
14M15	Grassmannians, Schubert varieties, flag manifolds
14C17	Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry
14H37	Automorphisms of curves

Keywords:

section of the Grassmannian; curve as intersection; restriction of an automorphism

Citations:

Zbl 0768.14014