## The Michigan Mathematical Journal

### Effective bounds on singular surfaces in positive characteristic

Jakub Witaszek

#### Article information

Source
Michigan Math. J., Volume 66, Issue 2 (2017), 367-388.

Dates
First available in Project Euclid: 6 April 2017

https://projecteuclid.org/euclid.mmj/1491465684

Digital Object Identifier
doi:10.1307/mmj/1491465684

Mathematical Reviews number (MathSciNet)
MR3657223

Zentralblatt MATH identifier
06750564

#### Citation

Witaszek, Jakub. Effective bounds on singular surfaces in positive characteristic. Michigan Math. J. 66 (2017), no. 2, 367--388. doi:10.1307/mmj/1491465684. https://projecteuclid.org/euclid.mmj/1491465684

#### References

• [1] V. Alexeev and S. Mori, Bounding singular surfaces of general type, Algebra, arithmetic and geometry with applications, pp. 143–174, Springer, Berlin, 2004.
• [2] P. Cascini, H. Tanaka, and J. Witaszek, On log del Pezzo surfaces in large characteristic, Compos. Math. 153 (2017), no. 4, 820–850.
• [3] P. Cascini, H. Tanaka, and C. Xu, On base point freeness in positive characteristic, Ann. Sci. Éc. Norm. Supér. 48 (2015), no. 5, 1239–1272.
• [4] G. Di Cerbo and A. Fanelli, Effective Matsusaka’s theorem for surfaces in characteristic $p$, Algebra Number Theory 9 (2015), no. 6, 1453–1475.
• [5] C. D. Hacon and C. Xu, On the three dimensional minimal model program in positive characteristic, J. Amer. Math. Soc. 28 (2015), no. 3, 711–744.
• [6] N. Hara, Classification of two-dimensional F-regular and F-pure singularities, Adv. Math. 133 (1998), no. 1, 33–53.
• [7] N. Hara, A characteristic $p$ analog of multiplier ideals and applications, Comm. Algebra 33 (2005), no. 10, 3375–3388.
• [8] N. Hara and K.-I. Watanabe, F-regular and F-pure rings vs. log terminal and log canonical singularities, J. Algebraic Geom. 11 (2002), no. 2, 363–392.
• [9] C. Jiang, Bounding the volumes of singular weak log del Pezzo surfaces, Internat. J. Math. 24 (2013), no. 13, 1350110.
• [10] D. S. Keeler, Fujita’s conjecture and Frobenius amplitude, Amer. J. Math. 130 (2008), no. 5, 1327–1336.
• [11] J. Kollár, Toward moduli of singular varieties, Compos. Math. 56 (1985), no. 3, 369–398.
• [12] J. Kollár, Effective base point freeness, Math. Ann. 296 (1993), no. 4, 595–605.
• [13] J. Kollár, Rational curves on algebraic varieties, Ergeb. Math. Grenzgeb. (3). Folge. A Ser. Mod. Surv. Math. [Results Math. Relat. Areas. 3rd Ser. A Ser. Mod. Surv. Math.], 32, Springer-Verlag, Berlin, 1996.
• [14] J. Kollár and S. Mori, Birational geometry of algebraic varieties, Cambridge Tracts in Math., 134, Cambridge University Press, Cambridge, 1998, With the collaboration of C. H. Clemens and A. Corti; Translated from the 1998 Japanese original.
• [15] C.-J. Lai, Bounding volumes of singular Fano threefolds, Nagoya Math. J. 224 (2016), no. 1, 37–73.
• [16] A. Langer, Adjoint linear systems on normal log surfaces, Compos. Math. 129 (2001), no. 1, 47–66.
• [17] R. Lazarsfeld, Positivity in algebraic geometry. I: classical setting: line bundles and linear series, Ergeb. Math. Grenzgeb. (3). Folge. A Ser. Mod. Surv. Math. [Results Math. Relat. Areas. 3rd Ser. A Ser. Mod. Surv. Math.], 48, Springer-Verlag, Berlin, 2004.
• [18] R. Lazarsfeld, Positivity in algebraic geometry. II: positivity for vector bundles, and multiplier ideals, Ergeb. Math. Grenzgeb. (3). Folge. A Ser. Mod. Surv. Math. [Results Math. Relat. Areas. 3rd Ser. A Ser. Mod. Surv. Math.], 49, Springer-Verlag, Berlin, 2004.
• [19] D. Martinelli, Y. Nakamura, and J. Witaszek, On the basepoint-free theorem for log canonical threefolds over the algebraic closure of a finite field, Algebra Number Theory 9 (2015), no. 3, 725–747.
• [20] T. Matsusaka, Polarized varieties with a given Hilbert polynomial, Amer. J. Math. 94 (1972), 1027–1077.
• [21] I. Reider, Vector bundles of rank $2$ and linear systems on algebraic surfaces, Ann. of Math. (2) 127 (1988), no. 2, 309–316.
• [22] F. Sakai, Reider–Serrano’s method on normal surfaces, Algebraic geometry (L’Aquila, 1988), Lecture Notes in Math., 1417, pp. 301–319, Springer, Berlin, 1990.
• [23] K. Schwede, F-adjunction, Algebra Number Theory 3 (2009), no. 8, 907–950.
• [24] K. Schwede, A canonical linear system associated to adjoint divisors in characteristic $p>0$, J. Reine Angew. Math. 696 (2014), 69–87.
• [25] K. Schwede, F-singularities and Frobenius splitting notes, http://www.math.utah.edu/~schwede/frob/RunningTotal.pdf.
• [26] K. Schwede and K. Tucker, A survey of test ideals, Progress in commutative algebra 2: closures, finiteness and factorization, pp. 39–99, de Gruyter, Berlin, 2012.
• [27] F. Serrano, Extension of morphisms defined on a divisor, Math. Ann. 277 (1987), no. 3, 395–413.
• [28] N. I. Shepherd-Barron, Unstable vector bundles and linear systems on surfaces in characteristic $p$, Invent. Math. 106 (1991), no. 2, 243–262.
• [29] K. E. Smith, Fujita’s freeness conjecture in terms of local cohomology, J. Algebraic Geom. 6 (1997), no. 3, 417–429.
• [30] K. E. Smith, A tight closure proof of Fujita’s freeness conjecture for very ample line bundles, Math. Ann. 317 (2000), no. 2, 285–293.
• [31] S. Takagi, Fujita’s approximation theorem in positive characteristics, J. Math. Kyoto Univ. 47 (2007), no. 1, 179–202.
• [32] H. Tanaka, Minimal models and abundance for positive characteristic log surfaces, Nagoya Math. J. 216 (2014), 1–70.
• [33] H. Tanaka, The trace map of Frobenius and extending sections for threefolds, Michigan Math. J. 64 (2015), no. 2, 227–261.
• [34] H. Tanaka, The X-method for KLT surfaces in positive characteristic, J. Algebraic Geom. 24 (2015), no. 4, 605–628.
• [35] H. Terakawa, The $d$-very ampleness on a projective surface in positive characteristic, Pacific J. Math. 187 (1999), no. 1, 187–199.
• [36] A. Valery, Boundedness and ${K}^{2}$ for log surfaces, Internat. J. Math. 5 (1994), no. 6, 779–810.