The Picard rank conjecture for the Hurwitz spaces of degree up to five (Q2018358)

After having fixed two positive integers \(g\geq 3\) and \(d\), one can consider the functor of isomorphism classes of smooth, proper and connected complex curves of genus \(g\), admitting a finite morphism of degree \(d\) to \(\mathbb{P}_{\mathbb{C}}^1\), whose branch divisor is supported at \(2g+2d-2\) distinct points. This moduli problem is known to have a coarse moduli space, denoted by \(\mathcal{H}_{d,g}\), which is a normal, \(\mathbb{Q}\)-factorial and irreducible quasi-projective complex variety of dimension \(2g+2d-5\). The Picard rank conjecture predicts that \(\mathrm{Pic}(\mathcal{H}_{d,g})\otimes \mathbb{Q}=0\). One consequence of this conjecture is the expectation that a certain partial compactification \(\widetilde{\mathcal{H}_{d,g}}\) of \(\mathcal{H}_{d,g}\), obtained by allowing nodal, irreducible curves and non simply branches, should have rational Picard group generated by boundary classes. It is known that the boundary classes can be expressed in terms of ``tautological classes''. In particular the expectation is that \(\mathrm{Pic}(\widetilde{\mathcal{H}_{d,g}})\otimes \mathbb{Q}\) is generated by tautological classes. The validity of the conjecture was known before the paper under review for \(d=2,3\) and for large \(d\), namely \(d>2g-2\). In the paper under review the authors prove this conjecture under the assumption that \(3\leq d\leq 5\) (Theorem A). The strategy of the proof consists in first showing that for \(d\geq 3\) (resp. \(d\geq 4\)) there are at least two (resp. at least three) divisorial components supported on \(\widetilde{\mathcal{H}_{d,g}}\setminus\mathcal{H}_{d,g}\) whose classes are linearly independent in \(\mathrm{Pic}( \widetilde{\mathcal{H}_{d,g}})\otimes \mathbb{Q}\) (Proposition 2.15). The most delicate part is then to show that \(\mathrm{rk}(\mathrm{Pic}( \widetilde{\mathcal{H}_{d,g}})\otimes \mathbb{Q})\leq 2\) (resp. \(\leq 3\)) for \(d=3\) (resp. for \(d=4,5\)). In order to produce these upper bounds the authors find a suitable open \(U\subset \widetilde{\mathcal{H}_{d,g}}\) which can be expressed as successive quotients of an open in a projective space by the action of explicit linear algebraic groups and whose number of divisorial components in \(\widetilde{\mathcal{H}_{d,g}}\setminus U\) can be explicitly computed. The authors associate to each smooth curve \(C\) of genus \(g\), equipped with a degree \(d\) morphism \(C\rightarrow \mathbb{P}_{\mathbb{C}}^1\) an embedding of \(C\) into the projectification of the associated Tschirnhausen bundle over \(\mathbb{P}_{\mathbb{C}}^1\). A resolution of the structure sheaf of the curve via this embedding is computed by using theorem 2.1 in [\textit{G. Casnati} and \textit{T. Ekedahl}, J. Algebr. Geom. 5, No. 3, 439--460 (1996; Zbl 0866.14009)]. The open \(U\subset \widetilde{\mathcal{H}_{d,g}}\) is obtained via a suitable mixing (depending on \(d\)) of the loci of curves where the associated Tschirnausen bundle and the first bundle in the Casnati-Ekedahl resolution are most generic. The loci are compared with certain Severi varieties as considered in [\textit{A. Ohbuchi}, J. Math., Tokushima Univ. 31, 7--10 (1997; Zbl 0938.14011)]. These Severi varieties are defined as follows. Fix \(m\) a positive integer, consider the Hirzebruch surface \(\mathbb{F}_m\) and \(\tau\subset \mathbb{F}_m\) a section with self-intersection equal to \(m\). Define \(\mathcal{V}_g(\mathbb{F}_m,d\tau)\) as the closure in the linear series \(|d\tau|\) of the locus \(\mathcal{U}_g(\mathbb{F}_m,d\tau)\) parametrizing irreducible, nodal curves of genus \(g\) in \(|d\tau|\). Ohbuchi gives a way to produce from a smooth curve of genus \(g\) equipped with a degree \(d\) morphism to \(\mathbb{P}_{\mathbb{C}}^1\) and a positive integer \(m\) a point in the Severi variety \(\mathcal{V}_g(\mathbb{F}_m,d\tau)\). \noindent As consequence of this association the authors can show that for any \(d\) and \(m\) larger than an explicit number (depending on \(d\) and \(g\)) the Picard rank conjecture is equivalent to \(\mathrm{Pic}(\mathcal{U}_g(\mathbb{F}_m,d\tau))\otimes\mathbb{Q}=0\) (Theorem B).