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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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