Moduli of ${\mathbb {Q}}$-Gorenstein pairs and applications
Authors:
Stefano Filipazzi and Giovanni Inchiostro
Journal:
J. Algebraic Geom. 33 (2024), 347-399
DOI:
https://doi.org/10.1090/jag/823
Published electronically:
January 2, 2024
Full-text PDF
Abstract |
References |
Additional Information
Abstract: We develop a framework to construct moduli spaces of ${\mathbb {Q}}$-Gorenstein pairs. To do so, we fix certain invariants; these choices are encoded in the notion of ${\mathbb {Q}}$-stable pair. We show that these choices give a proper moduli space with projective coarse moduli space and they prevent some pathologies of the moduli space of stable pairs when the coefficients are smaller than $\frac {1}{2}$. Lastly, we apply this machinery to provide an alternative proof of the projectivity of the moduli space of stable pairs.
References
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- János Kollár, Moduli of varieties of general type, Handbook of moduli. Vol. II, Adv. Lect. Math. (ALM), vol. 25, Int. Press, Somerville, MA, 2013, pp. 131–157. MR 3184176
- János Kollár, Singularities of the minimal model program, Cambridge Tracts in Mathematics, vol. 200, Cambridge University Press, Cambridge, 2013. With a collaboration of Sándor Kovács. MR 3057950, DOI 10.1017/CBO9781139547895
- János Kollár, Semi-normal log centres and deformations of pairs, Proc. Edinb. Math. Soc. (2) 57 (2014), no. 1, 191–199. MR 3165020, DOI 10.1017/S0013091513000801
- János Kollár, Families of divisors, arXiv:1910.00937 (2019).
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- János Kollár, Projectivity of complete moduli, J. Differential Geom. 32 (1990), no. 1, 235–268. MR 1064874
- Sándor J. Kovács and Zsolt Patakfalvi, Projectivity of the moduli space of stable log-varieties and subadditivity of log-Kodaira dimension, J. Amer. Math. Soc. 30 (2017), no. 4, 959–1021. MR 3671934, DOI 10.1090/jams/871
- J. Kollár and N. I. Shepherd-Barron, Threefolds and deformations of surface singularities, Invent. Math. 91 (1988), no. 2, 299–338. MR 922803, DOI 10.1007/BF01389370
- Gérard Laumon and Laurent Moret-Bailly, Champs algébriques, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 39, Springer-Verlag, Berlin, 2000 (French). MR 1771927, DOI 10.1007/978-3-540-24899-6
- Yongnam Lee and Noboru Nakayama, Grothendieck duality and $\Bbb Q$-Gorenstein morphisms, Publ. Res. Inst. Math. Sci. 54 (2018), no. 3, 517–648. MR 3834279, DOI 10.4171/PRIMS/54-3-3
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- Joaquín Moraga, Extracting non-canonical places, Adv. Math. 375 (2020), 107415, 12. MR 4170231, DOI 10.1016/j.aim.2020.107415
- David Mumford, Abelian varieties, Tata Institute of Fundamental Research Studies in Mathematics, vol. 5, Published for the Tata Institute of Fundamental Research, Bombay by Oxford University Press, London, 1970. MR 282985
- The Stacks Project authors, The Stacks Project, 2021.
- Eckart Viehweg, Quasi-projective moduli for polarized manifolds, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 30, Springer-Verlag, Berlin, 1995. MR 1368632, DOI 10.1007/978-3-642-79745-3
References
- Kenneth Ascher and Dori Bejleri, Moduli of weighted stable elliptic surfaces and invariance of log plurigenera, Proc. Lond. Math. Soc. (3) 122 (2021), no. 5, 617–677. With an appendix by Giovanni Inchiostro. MR 4258169, DOI 10.1112/plms.12387
- Kenneth Ascher, Dori Bejleri, Giovanni Inchiostro, and Zsolt Patakfalvi, Wall crossing for moduli of stable log varieties, Ann. Math. (2023), To appear.
- Enrico Arbarello, Maurizio Cornalba, and Phillip A. Griffiths, Geometry of algebraic curves. Volume II, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 268, Springer, Heidelberg, 2011. With a contribution by Joseph Daniel Harris. MR 2807457, DOI 10.1007/978-3-540-69392-5
- Dan Abramovich and Brendan Hassett, Stable varieties with a twist, Classification of algebraic varieties, EMS Ser. Congr. Rep., Eur. Math. Soc., Zürich, 2011, pp. 1–38. MR 2779465, DOI 10.4171/007-1/1
- Valery Alexeev, Higher-dimensional analogues of stable curves, International Congress of Mathematicians. Vol. II, Eur. Math. Soc., Zürich, 2006, pp. 515–536. MR 2275608
- Valery Alexeev, Moduli of weighted hyperplane arrangements, Advanced Courses in Mathematics. CRM Barcelona, Birkhäuser/Springer, Basel, 2015. Edited by Gilberto Bini, Martí Lahoz, Emanuele Macrìand Paolo Stellari. MR 3380944, DOI 10.1007/978-3-0348-0915-3
- Valery Alexeev, Boundedness and $K^2$ for log surfaces, Internat. J. Math. 5 (1994), no. 6, 779–810. MR 1298994, DOI 10.1142/S0129167X94000395
- Valery Alexeev, Moduli spaces $M_{g,n}(W)$ for surfaces, Higher-dimensional complex varieties (Trento, 1994) de Gruyter, Berlin, 1996, pp. 1–22. MR 1463171, DOI 10.1006/jcat.1996.0357
- Jarod Alper, Introduction to stacks and moduli (2021).
- Caucher Birkar, Boundedness and volume of generalised pairs, arXiv:2103.14935 (2021).
- Caucher Birkar, Moduli of algebraic varieties, arXiv:2211.11237 (2022).
- Osamu Fujino, Non-vanishing theorem for log canonical pairs, J. Algebraic Geom. 20 (2011), no. 4, 771–783. MR 2819675, DOI 10.1090/S1056-3911-2010-00558-9
- Kento Fujita, Semi-terminal modifications of demi-normal pairs, Int. Math. Res. Not. IMRN 24 (2015), 13653–13668. MR 3436159, DOI 10.1093/imrn/rnv114
- Osamu Fujino, Semipositivity theorems for moduli problems, Ann. of Math. (2) 187 (2018), no. 3, 639–665. MR 3779955, DOI 10.4007/annals.2018.187.3.1
- A. Grothendieck, Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. I, Inst. Hautes Études Sci. Publ. Math. 20 (1964), 259 (French). MR 173675
- Brendan Hassett, Moduli spaces of weighted pointed stable curves, Adv. Math. 173 (2003), no. 2, 316–352. MR 1957831, DOI 10.1016/S0001-8708(02)00058-0
- Brendan Hassett and Sándor J. Kovács, Reflexive pull-backs and base extension, J. Algebraic Geom. 13 (2004), no. 2, 233–247. MR 2047697, DOI 10.1090/S1056-3911-03-00331-X
- Christopher D. Hacon, James McKernan, and Chenyang Xu, ACC for log canonical thresholds, Ann. of Math. (2) 180 (2014), no. 2, 523–571. MR 3224718
- Christopher D. Hacon, James McKernan, and Chenyang Xu, Boundedness of moduli of varieties of general type, J. Eur. Math. Soc. (JEMS) 20 (2018), no. 4, 865–901. MR 3779687
- Christopher D. Hacon, James McKernan, and Chenyang Xu, Boundedness of varieties of log general type, Algebraic geometry: Salt Lake City 2015, 2018, pp. 309–348. MR 3821154
- Christopher D. Hacon and Chenyang Xu, Existence of log canonical closures, Invent. Math. 192 (2013), no. 1, 161–195. MR 3032329
- Christopher D. Hacon and Chenyang Xu, On finiteness of B-representations and semi-log canonical abundance, Minimal models and extremal rays (Kyoto, 2011), 2016, pp. 361–377. MR 3618266
- Giovanni Inchiostro, Moduli of Weierstrass fibrations with marked section, Adv. Math. 375 (2020), 107374, 57. MR 4137071, DOI 10.1016/j.aim.2020.107374
- János Kollár and Shigefumi Mori, 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. MR 1658959, DOI 10.1017/CBO9780511662560
- János Kollár, Hulls and husks, arXiv:0805.0576 (2008).
- János Kollár, Moduli of varieties of general type, Handbook of moduli. Vol. II, Adv. Lect. Math. (ALM), vol. 25, Int. Press, Somerville, MA, 2013, pp. 131–157. MR 3184176
- János Kollár, Singularities of the minimal model program, Cambridge Tracts in Mathematics, vol. 200, Cambridge University Press, Cambridge, 2013. With a collaboration of Sándor Kovács. MR 3057950, DOI 10.1017/CBO9781139547895
- János Kollár, Semi-normal log centres and deformations of pairs, Proc. Edinb. Math. Soc. (2) 57 (2014), no. 1, 191–199. MR 3165020, DOI 10.1017/S0013091513000801
- János Kollár, Families of divisors, arXiv:1910.00937 (2019).
- János Kollár, Families of varieties of general type, Cambridge Tracts in Mathematics, vol. 231, Cambridge University Press, Cambridge, 2023. With the collaboration of Klaus Altmann and Sándor J. Kovács. MR 4566297
- János Kollár, Projectivity of complete moduli, J. Differential Geom. 32 (1990), no. 1, 235–268. MR 1064874
- Sándor J. Kovács and Zsolt Patakfalvi, Projectivity of the moduli space of stable log-varieties and subadditivity of log-Kodaira dimension, J. Amer. Math. Soc. 30 (2017), no. 4, 959–1021. MR 3671934, DOI 10.1090/jams/871
- János Kollár and Nicholas I. Shepherd-Barron, Threefolds and deformations of surface singularities, Invent. Math. 91 (1988), no. 2, 299–338. MR 922803
- Gérard Laumon and Laurent Moret-Bailly, Champs algébriques, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 39, Springer-Verlag, Berlin, 2000 (French). MR 1771927
- Yongnam Lee and Noboru Nakayama, Grothendieck duality and $\mathbb {Q}$-Gorenstein morphisms, Publ. Res. Inst. Math. Sci. 54 (2018), no. 3, 517–648. MR 3834279, DOI 10.4171/PRIMS/54-3-3
- Vladimir Lazić and Nikolaos Tsakanikas, On the existence of minimal models for log canonical pairs, Publ. Res. Inst. Math. Sci. 58 (2022), no. 2, 311–339. MR 4416583, DOI 10.4171/prims/58-2-3
- Joaquín Moraga, Extracting non-canonical places, Adv. Math. 375 (2020), 107415, 12. MR 4170231, DOI 10.1016/j.aim.2020.107415
- David Mumford, Abelian varieties, Tata Institute of Fundamental Research Studies in Mathematics, vol. 5, Published for the Tata Institute of Fundamental Research, Bombay by Oxford University Press, London, 1970. MR 0282985
- The Stacks Project authors, The Stacks Project, 2021.
- Eckart Viehweg, Quasi-projective moduli for polarized manifolds, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 30, Springer-Verlag, Berlin, 1995. MR 1368632, DOI 10.1007/978-3-642-79745-3
Additional Information
Stefano Filipazzi
Affiliation:
EPFL, SB MATH CAG, MA C3 625 (Bâtiment MA), Station 8, CH-1015 Lausanne, Switzerland
MR Author ID:
1272161
ORCID:
setImmediate$0.7793210018488271$12
Email:
stefano.filipazzi@epfl.ch
Giovanni Inchiostro
Affiliation:
Department of Mathematics, University of Washington, Box 354350, Seattle, Washington
MR Author ID:
1334068
ORCID:
0000-0002-9989-5937
Email:
ginchios@uw.edu
Received by editor(s):
December 19, 2021
Received by editor(s) in revised form:
February 18, 2022, and May 18, 2023
Published electronically:
January 2, 2024
Additional Notes:
The first author was partially supported by ERC starting grant #804334
Article copyright:
© Copyright 2024
University Press, Inc.
