Sharp slope bounds for sweeping families of trigonal curves (Q2510093)

Let \(\overline{M}_g\) be the moduli space of Deligne-Mumford stable curves of genus \(g\). Denote by \(\lambda\) the first Chern class of the Hodge bundle and by \(\delta\) the total boundary class on \(\overline{M}_g\). For a curve \(B\) in \(\overline{M}_g\), i.e., \(B\) is a one-dimensional family of stable genus \(g\) curves, define its slope by \(\deg_B (\delta) / \deg_B (\lambda)\). For a subvariety \(X\subset \overline{M}_g\), one can ask what is the highest \(s\) such that \(X\) is covered by curves of slope \(s\)? An answer to this question can help us better understand the birational geometry of \(\overline{M}_g\), e.g., if \(X\) can be covered by curves of slope \(s_0\), then it must be in the base locus of the linear system \(|s\lambda - \delta|\) for \(s < s_0\). An important type of subvarieties of \(\overline{M}_g\) is given by the closure of the locus of \(d\)-gonal curves, i.e., curves admitting a degree \(d\) branched cover of \(\mathbb P^1\). For the beginning case \(d = 2\), i.e., the locus of hyperelliptic curves, \textit{M. Cornalba} and \textit{J. Harris} [Ann. Sci. Éc. Norm. Supér. (4) 21, No. 3, 455--475 (1988; Zbl 0674.14006)] obtained the sharp slope bound \(8 + 4/g\). For the next case \(d=3\), i.e., the locus of trigonal curves, \textit{Z. E. Stankova-Frenkel} [J. Algebr. Geom. 9, No. 4, 607--662 (2000; Zbl 1001.14007)] obtained the sharp slope bound \(7 + 6/g\) if \(g\) is even. In the paper under review, the authors study trigonal curves and obtained the sharp slope bound \(7+20/(3g+1)\) if \(g\) is odd. They also reprove Stankova's bound for even \(g\), thus completely answering the slope question for the locus of trigonal curves. To prove their results, the authors explicitly construct curves with desired slopes that sweep out the locus of trigonal curves. Moreover, they construct effective divisors on the Hurwitz space of trigonal curves and compute their divisor classes to match with the slope bounds. For even \(g\), they consider the divisor given by the locus of trigonal curves that embed in an unbalanced scroll. For odd \(g\), the divisor is the locus of trigonal curves that embed in the Hirzebruch surface \(\mathbf{F}_1\) and are tangent to the directrix. In order to make the calculation feasible, some modern techniques such as twisted admissible covers and Grothendieck-Riemann-Roch for stacks are invoked and applied.