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Stable pairs with a twist and gluing morphisms for moduli of surfaces

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Abstract

We propose an alternative definition for families of stable pairs (X, D) over an arbitrary (possibly non-reduced) base in the case in which D is reduced, by replacing (X, D) with an appropriate orbifold pair \((\mathcal {X},\mathcal {D})\). This definition of a stable family ends up being equivalent to previous ones, but has the advantage of being more amenable to the tools of deformation theory. Adjunction for \((\mathcal {X},\mathcal {D})\) holds on the nose; there is no correction term coming from the different. This leads to the existence of functorial gluing morphisms for families of stable surfaces and functorial morphisms from \((n + 1)\) dimensional stable pairs to n dimensional polarized orbispaces. As an application, we study the deformation theory of some surface pairs.

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Notes

  1. In this setting the family of divisors is automatically flat (see Corollary 3.6).

  2. The volume of a line bundle on \(\mathcal {X}\) is defined to be the volume of the corresponding \(\mathbb {Q}\)-Cartier divisor on the coarse space.

  3. For properties of group quotients of DM stack, see [40].

  4. Such pushouts are often called pinchings or Ferrand pushouts.

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Acknowledgements

The authors would like to thank Dan Abramovich, Shamil Asgarli, Brendan Hassett, János Kollár, Sándor Kovács, Davesh Maulik, Jonathan Wise, and Chenyang Xu for many helpful conversations. We thank David Rydh for informing us of Theorem 5.12 and its proof. We also thank the referees for reading the draft carefully and giving insightful feedback. The first author is supported by an NSF Postdoctoral Fellowship DMS-1803124 and was partially supported by NSF Grant DMS-1759514. The second author is partially supported by NSF Grant DMS-1759514. The second author also thanks the UC Berkeley Department of Mathematics for their hospitality while part of this research was conducted.

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  1. Harvard University, Cambridge, MA, USA

    Dori Bejleri

  2. University of Washington, Seattle, WA, USA

    Giovanni Inchiostro

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  1. Dori Bejleri
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  2. Giovanni Inchiostro
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Correspondence to Dori Bejleri.

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Bejleri, D., Inchiostro, G. Stable pairs with a twist and gluing morphisms for moduli of surfaces. Sel. Math. New Ser. 27, 40 (2021). https://doi.org/10.1007/s00029-021-00661-2

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