Stable pairs with a twist and gluing morphisms for moduli of surfaces
Tóm tắt
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.
Tài liệu tham khảo
Ascher, K., Bejleri, D.: Compact Moduli of Elliptic K3 Surfaces (2019)
Abramovich, D., Brendan, H.: Stable varieties with a twist. Classif. Algebr. Var. 27, 1–38 (2011)
Altmann, K., Kollár, J.: The dualizing sheaf on first-order deformations of toric surface singularities. J. Reine Angew. Math. (Crelles J.) (2016)
Alexeev, V.: Boundedness and K2 for log surfaces. Int. J. Math. 5(6), 779–810 (1994)
Abramovich, D., Olsson, M., Vistoli, A.: Tame stacks in positive characteristic. Ann. Inst. Fourier (Grenoble) 58(4), 1057–1091 (2008)
Abramovich, D., Olsson, M., Vistoli, A.: Twisted stable maps to tame Artin stacks. J. Algebr. Geom. 20(3), 399–477 (2011)
Birkar, C., Cascini, P., Hacon, C.D., McKernan, J.: Existence of minimal models for varieties of log general type. J. Am. Math. Soc. 23(2), 405–468 (2010)
Bruns, W., Herzog, J.: Cohen-Macaulay Rings. Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge (1993)
Conrad, B.: Grothendieck Duality and Base Change. Lecture Notes in Mathematics, vol. 1750. Springer, Berlin (2000)
Deopurkar, D., Han, C.: Stable Log Surfaces, Admissible Covers, and Canonical Curves of Genus 4. arXiv:1807.08413 (2018)
Deligne, P., Mumford, D.: The irreducibility of the space of curves of given genus. Inst. Hautes Études Sci. Publ. Math. 36, 75–109 (1969)
Greb, D., Kebekus, S., Kovacs, S.J., Peternell, T.: Differential forms on log canonical spaces. Publ. Math. Inst. Hautes Études Sci. 114, 87–169 (2011)
Grothendieck, A.: Éíements de géomètrie algébrique (rédigés avec la collaboration de Jean Dieudonné): IV. étude locale des schémas et des morphismes de schémas, troisiéme partie. Inst. Hautes Études Sci. Publ. Math. 28 (1966)
Hacking, P.: Compact moduli of plane curves. Duke Math. J. 124(2), 213–257 (2004)
Hassett, B., Kovács, S.: Reflexive pull-backs and base extension. J. Algebr. Geom. 13(2), 233–247 (2004)
Hacon, C.D., McKernan, J., Xu, C.: Boundedness of moduli of varieties of general type. J. Eur. Math. Soc. (JEMS) 20(4), 865–901 (2018)
Hacon, C.D., Xu, C.: Existence of log canonical closures. Invent. Math. 192(1), 161–195 (2013)
Karu, K.: Minimal models and boundedness of stable varieties. J. Algebr. Geom. 9(1), 93–109 (2000)
Kollár, J., Kovács, S.: Deformations of log canonical and F-pure singularities. ArXiv e-prints (2018) Available at 1807.07417
Kollár, J., Mori, S.: Birational Geometry of Algebraic Varieties. Cambridge Tracts in Mathematics, vol. 134. Cambridge University Press, Cambridge (1998). (With the collaboration of C. H. Clemens and A. Corti, Translated from the 1998 Japanese original)
Knudsen, F.F.: The projectivity of the moduli space of stable curves. III. The line bundles on Mg,n, and a proof of the projectivity of Mg,n in characteristic 0. Math. Scand. 52(2), 200–212 (1983)
Kollár, J.: Hulls and Husks. arXiv:0805.0576 (2008)
Kollár, J.: Simultaneous normalization and algebra husks. Asian J. Math. 15(3), 437–449 (2011)
Kollár, J.: Moduli of varieties of general type. Handb. Moduli II, 131–157 (2013)
Kollár, J.: Singularities of the Minimal Model Program. Cambridge Tracts in Mathematics, vol. 200. Cambridge University Press, Cambridge (2013). (With a collaboration of Sándor Kovács)
Kollár, J.: Families of Varieties of General Type. https://web.math.princeton.edu/~kollar/ (2017)
Kollár, J.: Families of Divisors. arXiv:1910.00937 (2019)
Kollár, J.: Projectivity of complete moduli. J. Differ. Geom. 32(1), 235–268 (1990)
Kovács, S.: Rational Singularities. ArXiv e-prints (2017) Available at 1703.02269
Kovács, S., Patakfalvi, Z.: Projectivity of the moduli space of stable log-varieties and subadditivity of log-Kodaira dimension. J. Am. Math. Soc. 30(4), 959–1021 (2017)
Kollár, J., Shepherd-Barron, N.I.: Threefolds and deformations of surface singularities. Invent. Math. 91(2), 299–338 (1988)
Lieblich, M.: Remarks on the stack of coherent algebras. Int. Math. Res. Not. Article ID: 75273, 12 (2006)
Lee, Y., Nakayama, N.: Grothendieck duality and Q-Gorenstein morphisms. Publ. Res. Inst. Math. Sci. 54(3), 517–648 (2018)
Matsumura, H.: Commutative Ring Theory, Second. Cambridge Studies in Advanced Mathematics, vol. 8. Cambridge University Press, Cambridge (1989). (Translated from the Japanese by M. Reid)
Miranda, R.: The Basic Theory of Elliptic Surfaces. Dottorato di Ricerca in Matematica. [Doctorate in Mathematical Research], ETS Editrice, Pisa (1989)
Olsson, M.C.: Deformation theory of representable morphisms of algebraic stacks. Math. Z. 253(1), 25–62 (2006)
Olsson, M.C.: Hom-stacks and restriction of scalars. Duke Math. J. 134(1), 139–164 (2006)
Olsson, M.: Algebraic Spaces and Stacks. American Mathematical Society Colloquium Publications, vol. 62. American Mathematical Society, Providence (2016)
Patakfalvi, Z.: Fibered stable varieties. Trans. Am. Math. Soc. 368(3), 1837–1869 (2016)
Romagny, M.: Group actions on stacks and applications. Mich. Math. J. 53(1), 209–236 (2005)
Rydh, D.: Compactification of Tame Deligne-Mumford Stacks. Preprint (2014)
Schwede, S.: Obtaining Non-normal Varieties by Pushout. https://mathoverflow.net/q/186650
Włodarczyk, J.: Simple Hironaka resolution in characteristic zero. J. Am. Math. Soc. 18(4), 779–822 (2005)