On the monodromy group of the family of smooth quintic plane curves

Salter, Nick

Glasgow mathematical journal, Tome 67 (2025) no. 2, pp. 163-184

Résumé

We consider the space $\mathcal{P}_d$ of smooth complex projective plane curves of degree $d$. There is the tautological family of plane curves defined over $\mathcal{P}_d$, which has an associated monodromy representation $\rho _d: \pi _1(\mathcal{P}_d) \to \textrm{Mod}(\Sigma _g)$ into the mapping class group of the fiber. For $d \le 4$, classical algebraic geometry implies the surjectivity of $\rho _d$. For $d \ge 5$, the existence of a $(d-3)^{rd}$ root of the canonical bundle implies that $\rho _d$ cannot be surjective. The main result of this paper is that for $d = 5$, the image of $\rho _5$ is as large as possible, subject to this constraint. This requires combining the algebro-geometric work of Lönne with Johnson’s theory of the Torelli subgroup of $\textrm{Mod}(\Sigma _g)$.

Détail

DOI : 10.1017/S0017089524000284

Mots-clés : Plane curves, monodromy, mapping class group, r-spin structure