Curves and symmetric spaces. III: BN-special vs. 1-PS degeneration
The author continues his investigation of curves of genus \(7,\,8,\, 9\) which can be described as transversal linear sections of certain symmetric spaces. A curve \(C\) of genus \(g\) is \textit{Brill-Noether general} if \(h^0(\xi)h^0(K_C\xi^{-1})\le g\) holds, for every line bundle \(\xi\) on \(C\) with \(h^0(\xi)\ge 2\) and \(h^0(K_C\xi^{-1})\ge 2\). For \(g=7,8,9\) it was proved that a Brill-Noether general curve of genus \(g\) is isomorphic to a transversal linear section \(\Sigma_{2g-2}\cap H_1\cap\ldots\cap H_{23-2g}\), where \(\Sigma_{2g-2}=G/P\subset \mathbf{P}(V)\) is a symmetric flag manifold of dimension \(24-2g\) and \(V\) is a \(G\)-module (see [\textit{S. Mukai}, Conf. Proc. Lect. Notes Algebr. Geom. 1, 19--40 (1993; Zbl 0846.14030); Proc. Japan Acad., Ser. A 68, No. 1, 7--10 (1992; Zbl 0768.14014); Am. J. Math. 117, No. 6, 1627--1644 (1995; Zbl 0871.14025); Ann. Math. (2) 172, No. 3, 1539--1558 (2010; Zbl 1210.14034)]). The main theorem of this paper extends the above result to curves in a non-empty open subset of the irreducible divisor \(BNS_g \subset \textbf{M}_g\) formed by the curves which are \textit{not} Brill-Noether (here \(\textbf{M}_g\) denotes the moduli space of genus \(g\) curves). Theorem. For each \(g = 7, 8, 9\), there exists a one-parameter subgroup \(\lambda\in \mathbf{G}_m\subset SL(V)\) such that a curve \(C\) of genus \(g\) corresponding to a general point of \(BNS_g\) is isomorphic to a transversal linear section of the \(1-PS\) degeneration \[ [\Sigma'_{2g-2} \subset \mathbf{P}^{22-g} ] := \lim_{\lambda\to 0}[\Sigma^\lambda_{2g-2} \subset \mathbf{P}^{22-g} ]. \]
scientific article; zbMATH DE number 7606133