Theta-regularity and log-canonical threshold

  • Morten Øygarden
  • Sofia Tirabassi

Abstract

We show that an inequality, proven by Küronya-Pintye, which governs the behavior of the log-canonical threshold of an ideal over $\mathbb {P}^n$ and that of its Castelnuovo-Mumford regularity, can be applied to the setting of principally polarized abelian varieties by substituting the Castelnuovo-Mumford regularity with Θ-regularity of Pareschi-Popa.

References

Debarre, O., Fulton-Hansen and Barth-Lefschetz theorems for subvarieties of abelian varieties, J. Reine Angew. Math. 467 (1995), 187–197. https://doi.org/10.1515/crll.1995.467.187

Debarre, O., Théorèmes de connexité et variétés abéliennes, Amer. J. Math. 117 (1995), no. 3, 787–805. https://doi.org/10.2307/2375089

Debarre, O., On coverings of simple abelian varieties, Bull. Soc. Math. France 134 (2006), no. 2, 253–260. https://doi.org/10.24033/bsmf.2508

Ein, L. and Lazarsfeld, R., Singularities of theta divisors and the birational geometry of irregular varieties, J. Amer. Math. Soc. 10 (1997), no. 1, 243–258. https://doi.org/10.1090/S0894-0347-97-00223-3

Höring, A., $M$-regularity of the Fano surface, C. R. Math. Acad. Sci. Paris 344 (2007), no. 11, 691–696. https://doi.org/10.1016/j.crma.2007.04.008

Küronya, A. and Pintye, N., Castelnuovo-Mumford regularity and log-canonical thresholds, preprint arXiv:1312.7778 [math.AG], 2013.

Lazarsfeld, R., Positivity in algebraic geometry. I: Classical setting: line bundles and linear series, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, vol. 48, Springer-Verlag, Berlin, 2004. https://doi.org/10.1007/978-3-642-18808-4

Lazarsfeld, R., Positivity in algebraic geometry. II: positivity for vector bundles, and multiplier ideals, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, vol. 49, Springer-Verlag, Berlin, 2004. https://doi.org/10.1007/978-3-642-18810-7

Lombardi, L. and Niu, W., Theta-regularity of curves and Brill-Noether loci, Math. Res. Lett. 23 (2016), no. 6, 1761–1787. https://doi.org/10.4310/MRL.2016.v23.n6.a9

Pareschi, G., Basic results on irregular varieties via Fourier-Mukai methods, in “Current developments in algebraic geometry”, Math. Sci. Res. Inst. Publ., vol. 59, Cambridge Univ. Press, Cambridge, 2012, pp. 379–403.

Pareschi, G. and Popa, M., Regularity on abelian varieties. I, J. Amer. Math. Soc. 16 (2003), no. 2, 285–302. https://doi.org/10.1090/S0894-0347-02-00414-9

Pareschi, G. and Popa, M., Regularity on abelian varieties. II. Basic results on linear series and defining equations, J. Algebraic Geom. 13 (2004), no. 1, 167–193. https://doi.org/10.1090/S1056-3911-03-00345-X

Pareschi, G. and Popa, M., Castelnuovo theory and the geometric Schottky problem, J. Reine Angew. Math. 615 (2008), 25–44. https://doi.org/10.1515/CRELLE.2008.008

Pareschi, G. and Popa, M., Generic vanishing and minimal cohomology classes on abelian varieties, Math. Ann. 340 (2008), no. 1, 209–222. https://doi.org/10.1007/s00208-007-0146-7

Pareschi, G. and Popa, M., $M$-regularity and the Fourier-Mukai transform, Pure Appl. Math. Q. 4 (2008), no. 3, Special Issue: In honor of Fedor Bogomolov. Part 2, 587–611. https://doi.org/10.4310/PAMQ.2008.v4.n3.a1

Pareschi, G. and Popa, M., GV-sheaves, Fourier-Mukai transform, and generic vanishing, Amer. J. Math. 133 (2011), no. 1, 235–271. https://doi.org/10.1353/ajm.2011.0000

Pareschi, G. and Popa, M., Regularity on abelian varieties III: relationship with generic vanishing and applications, in “Grassmannians, moduli spaces and vector bundles”, Clay Math. Proc., vol. 14, Amer. Math. Soc., Providence, RI, 2011, pp. 141–167.
Published
2020-03-29
How to Cite
Øygarden, M., & Tirabassi, S. (2020). Theta-regularity and log-canonical threshold. MATHEMATICA SCANDINAVICA, 126(1), 73-81. https://doi.org/10.7146/math.scand.a-115971
Section
Articles