## Algebra & Number Theory

### A fibered power theorem for pairs of log general type

#### Abstract

Let $f : (X,D) → B$ be a stable family with log canonical general fiber. We prove that, after a birational modification of the base $B˜ → B$, there is a morphism from a high fibered power of the family to a pair of log general type. If in addition the general fiber is openly canonical, then there is a morphism from a high fibered power of the original family to a pair openly of log general type.

#### Article information

Source
Algebra Number Theory, Volume 10, Number 7 (2016), 1581-1600.

Dates
Revised: 30 June 2016
Accepted: 31 July 2016
First available in Project Euclid: 16 November 2017

https://projecteuclid.org/euclid.ant/1510842569

Digital Object Identifier
doi:10.2140/ant.2016.10.1581

Mathematical Reviews number (MathSciNet)
MR3554241

Zentralblatt MATH identifier
1376.14035

#### Citation

Ascher, Kenneth; Turchet, Amos. A fibered power theorem for pairs of log general type. Algebra Number Theory 10 (2016), no. 7, 1581--1600. doi:10.2140/ant.2016.10.1581. https://projecteuclid.org/euclid.ant/1510842569

#### References

• D. Abramovich, “A high fibered power of a family of varieties of general type dominates a variety of general type”, Invent. Math. 128:3 (1997), 481–494.
• D. Abramovich, “Uniformity of stably integral points on elliptic curves”, Invent. Math. 127:2 (1997), 307–317.
• D. Abramovich and B. Hassett, “Stable varieties with a twist”, pp. 1–38 in Classification of algebraic varieties, European Mathematical Society, Zürich, 2011.
• D. Abramovich and K. Matsuki, “Uniformity of stably integral points on principally polarized abelian varieties of dimension $\le2$”, Israel J. Math. 121 (2001), 351–380.
• D. Abramovich and J. F. Voloch, “Lang's conjectures, fibered powers, and uniformity”, New York J. Math. 2 (1996), 20–34, electronic.
• B. Bhatt, W. Ho, Z. Patakfalvi, and C. Schnell, “Moduli of products of stable varieties”, Compos. Math. 149:12 (2013), 2036–2070.
• L. Caporaso, J. Harris, and B. Mazur, “Uniformity of rational points”, J. Amer. Math. Soc. 10:1 (1997), 1–35.
• O. Fujino, “Semipositivity theorems for moduli problems”, preprint, 2012.
• R. Hartshorne, “Stable reflexive sheaves”, Math. Ann. 254:2 (1980), 121–176.
• B. Hassett, “Correlation for surfaces of general type”, Duke Math. J. 85:1 (1996), 95–107.
• B. Hassett and S. J. Kovács, “Reflexive pull-backs and base extension”, J. Algebraic Geom. 13:2 (2004), 233–247.
• H. Hironaka, “Idealistic exponents of singularity”, pp. 52–125 in Algebraic geometry (J. J. Sylvester Symposium (1976)), edited by J.-I. Igusa, Johns Hopkins University Press, Baltimore, MD, 1977.
• K. Karu, Semistable reduction in characteristic 0, Ph.D. thesis, Boston University, 1999, hook http://search.proquest.com/docview/304512245/ \posturlhook.
• J. Kollár, “Book on moduli of surfaces”, book in progress, 2010, hook https://web.math.princeton.edu/~kollar/ \posturlhook.
• J. Kollár, Singularities of the minimal model program, Cambridge Tracts in Mathematics 200, Cambridge University Press, 2013.
• J. Kollár and S. J. Kovács, “Log canonical singularities are Du Bois”, J. Amer. Math. Soc. 23:3 (2010), 791–813.
• J. Kollár and S. Mori, Birational geometry of algebraic varieties, Cambridge Tracts in Mathematics 134, Cambridge University Press, 1998.
• S. J. Kovács and Z. Patakfalvi, “Projectivity of the moduli space of stable log-varieties and subadditvity of log-Kodaira dimension”, preprint, 2015.
• P. L. Pacelli, “Uniform boundedness for rational points”, Duke Math. J. 88:1 (1997), 77–102.
• P. L. Pacelli, “Uniform bounds for stably integral points on elliptic curves”, Proc. Amer. Math. Soc. 127:9 (1999), 2535–2546.
• Z. Patakfalvi, “Fibered stable varieties”, Trans. Amer. Math. Soc. 368:3 (2016), 1837–1869.
• Z. Patakfalvi and C. Xu, “Ampleness of the CM line bundle on the moduli space of canonically polarized varieties”, preprint, 2015.