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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.
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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