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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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