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Positivity of divisor classes on the strata of differentials. (English) Zbl 1440.14028
The author proves results for families of stable \(k\)-differentials analogous to those of M. Cornalba and J. Harris [Ann. Sci. Éc. Norm. Supér. (4) 21, No. 3, 455–475 (1988; Zbl 0674.14006)] for families of stable curves. Fixing a genus \(g\), \(k>0\), and an \(n\)-tuple \(\mu = (m_1, m_2, \dots, m_n)\) of integers with \(\sum m_i = k(2g-2)\), let \(\mathcal H^k (\mu)\) be the space of tuples \((C, \xi, p_1, \dots, p_n)\) where \(p_i\) are distinct points on a smooth connected curve \(C\) of genus genus \(g\) and \(\xi \in H^0(K^{\otimes k}_C)\) is a section with \((\xi) = \sum m_i p_i\). Let \(\mathcal P^k (\mu)\) be the corresponding projectivization. M. Bainbridge et al. defined the incidence variety compactification \(\mathcal P^k (\mu) \subset \overline {\mathcal P}^k (\mu)\) and completely described the boundary [Duke Math. J. 167, 2347–2416 (2018; Zbl 1403.14058); Algebr. Geom. 6, 196–233 (2019; Zbl 1440.14148)]. The idea is to embed \(\mathcal P^k (\mu)\) into a twisted \(k\)th Hodge bundle, pull back to \(\mathcal M_{g,n}\), extend to \(\overline{\mathcal M}_{g,n}\) and take the closure.
For the results, let \(\psi_i\) denote the cotangent bundle associated to \(p_i\), \(\psi = \sum \psi_i\) and \(\kappa = \pi_* c_1^2 (\omega)\) on \(\overline {\mathcal M}_{g,n}\). Use the same notation for the pullbacks to \(\overline {\mathcal P}^k (\mu)\). Also let \(\eta\) be the pullback to \(\overline {\mathcal P}^k (\mu)\) of \(\mathcal O (-1)\) arising from twisted \(k\)th Hodge bundle. These line bundles pull back to any family \(\pi: \mathcal C \to B\) of pointed stable \(k\)-differentials \((C,\xi, p_1, \dots, p_n)\) of type \(\mu\) modulo scaling over a complete integral curve \(B\). The main theorem states that \[ \deg_{\pi} (k(\kappa + \psi) - \eta) \geq 0 \] i.e. \(k(\kappa + \psi) - \eta\) is numerically effecitve on \(\overline {\mathcal P}^k (\mu)\). A more refined statement is given involving the individual \(\psi_i\). The author includes a simplified proof when the generic fiber of the family is smooth for easier reading. He deduces that \(a(\kappa + \psi) - b \eta\) is ample on \(\overline {\mathcal P}^k (\mu)\) if \(a > bk > 0\). The last section gives examples showing sharpness of the theorem.
MSC:
14C17 Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry
14H15 Families, moduli of curves (analytic)
14H10 Families, moduli of curves (algebraic)
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