Semistable sheaves with symmetric \(c_1\) on Del Pezzo surfaces of degree \(5\) and \(6\)
The author realizes for Del Pezzo surfaces of degree 5 and 6 the program begun by \textit{J. M. Drezet} and \textit{J. Le Potier} [Ann. Sci. Éc. Norm. Supér. (4) 18, 193--243 (1985; Zbl 0586.14007)], deepened by \textit{J. M. Drezet} [J. Reine Angew. Math. 380, 14--58 (1987; Zbl 0613.14013)] and extended by the author of the paper under review for \({\mathbb P}^1 \times {\mathbb P}^1\) [\textit{T. Abe}, Nagoya Math. J. 227, 86--159 (2017; Zbl 1415.14015)]. In this paper the following aims are realized: \begin{itemize} \item ``determine Chern classes of semistable sheaves with symmetric \(c_1\) on Del Pezzo surfaces of degree 5 and 6 (Theorem 4.8); \item define the height of the moduli space of semistable sheaves with symmetric \(c_1\) (Definition 4.9); \item show that the moduli space of height zero of semistable sheaves with symmetric \(c_1\) is isomorphic to a moduli space of representations of a certain quiver (Theorem 6.4)''. \end{itemize} It is interesting to note that the author explains why the case of Del Pezzo surfaces of degree 5 and 6 is a natural continuation of the cases \({\mathbb P}^2\) and \({\mathbb P}^1 \times {\mathbb P}^1\).
scientific article; zbMATH DE number 7371784
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