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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 \(P\).

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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