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

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
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