The unirationality of the Hurwitz schemes $\mathcal H_{10, 8}$ and $\mathcal H_{13, 7}$

Abstract

We show that the Hurwitz scheme \mathcal{H}_{g,d} parametrizing d -sheeted simply branched covers of the projective line by smooth curves of genus g , up to isomorphism, is unirational for (g,d)=(10,8) and (13,7) . The unirationality is settled by using liaison constructions in \mathbb{P}^1 \times \mathbb{P}^2 and \mathbb{P}^6 respectively, and through the explicit computation of single examples over a finite field.

Fano 3-folds from homogeneous vector bundles over Grassmannians

“…To conclude, it suffices to show the existence of a curve with the right genus and degree having the desired resolution. This task can be rather difficult, depending on the given invariants: on P 1 ×P 2 , different approaches can be adopted, such as liaison theory or the construction of the Hartshorne-Rao module of the curve, see, e.g., [25,26]. Our situation, however, is favourable, as the minors of a general matrix…”

On the unirationality of moduli spaces of pointed curves

“…The number η(g) (respectively, τ (g)) denotes the smallest number of points n for which the Kodaira dimension of M g,n is known to be non-negative (respectively, maximal). The table is based on contributions given by [3,4,6,9,10,25,27,34] for the unirationality; [1,5,10,18,19,27] for the uniruledness; [8,16,19,20,27,32] for η(g) and τ (g).…”

Fano 3-folds from homogeneous vector bundles over Grassmannians

“…By To conclude, it suffices to show the existence of a curve with the right genus and degree having the desired resolution. This task can be rather difficult, depending on the given invariants: on P 1 × P 2 , different approaches can be adopted, such as liaison theory or the construction of the Hartshorne-Rao module of the curve, see, e.g., [KT19,KT]. Our situation, however, is favourable, as the minors of a general matrix O(−1, −4) ⊕2 → O(0, −2) ⊕ O(−1, −3) ⊕2 generate the ideal of a smooth curve of maximal rank with the desired invariants.…”