The unirationality of Hurwitz spaces of 6-gonal curves of small genus (Q1946047)

The author shows the unirationality of the Hurwitz spaces \(\mathcal{H}(6,2g+10)\) for a series of small genera, namely \(5 \leq g \leq 28\) and \(g = 30,31,35,36,40,45\). In general, the Hurwitz spaces \(\mathcal{H}(d,w)\) parameterize degree \(d\) coverings from a smooth genus \(g\) curve to \(\mathbb{P}^1\) that are simply branched at \(w\) points, where the occurring parameters are related by the Hurwitz formula \(w = 2g + 2d - 2\). These spaces are always smooth and irreducible [\textit{W.~Fulton}, Ann. Math. (2) 90, 542--575 (1969; Zbl 0194.21901)]. They had been known to be unirational for \(d \leq 5\) and \(g \geq d-1\), for \(d=6\) and \(5 \leq g \leq 10\) or \(g = 12\), and for \(d = g = 7\) by work of \textit{E.~Arbarello} and \textit{M.~Cornalba} [Math. Ann. 256, 341--362 (1981; Zbl 0454.14023)], who constructed a suitable plane model of the curve, from which the \(g^1_d\) could be obtained as a projection, and which could be parameterized rationally. In contrast, the present paper uses an approach via liaison theory in \(\mathbb{P} := \mathbb{P}^1 \times \mathbb{P}^2\). Two curves in \(\mathbb{P}\) are said to be linked if their union is a complete intersection. A generic curve \(C \in \mathcal{H}(6,2g+10)\) can be embedded into \(\mathbb{P}\) via the given \(g^1_6\) and a base point free \(g^2_d\) of minimal degree. The author shows that the image can be linked in two steps to a union of rational curves, and that for the above values of \(g\) the process can be reversed. In order to show that the resulting map is dominant, the author exhibits a Macaulay2 program that constructs a smooth absolutely irreducible curve \(C \subseteq \mathbb{P}\) of bidegree \((6,d)\) over a finite field from a random initial datum. The construction can be regarded as the reduction of a computation over \(\mathbb{Q}\), where the curve stays smooth and absolutely irreducible by semi-continuity. On an open subset of the corresponding Hilbert scheme, the projection to \(\mathbb{P}^1\) is simply branched, so in this way one obtains a dominant map to \(\mathcal{H}(6,2g+10)\). Since all parameters in the construction can be chosen rationally, this shows that the Hurwitz space is unirational.