## Duke Mathematical Journal

### Chern slopes of surfaces of general type in positive characteristic

Giancarlo Urzúa

#### Abstract

Let $\mathbf{{k}}$ be an algebraically closed field of characteristic $p\gt 0$, and let $C$ be a nonsingular projective curve over $\mathbf{{k}}$. We prove that for any real number $x\geq2$, there are minimal surfaces of general type $X$ over $\mathbf{{k}}$ such that (a) $c_{1}^{2}(X)\gt 0$, $c_{2}(X)\gt 0$, (b) $\pi_{1}^{\acute{e}t}(X)\simeq\pi_{1}^{\acute{e}t}(C)$, and (c) $c_{1}^{2}(X)/c_{2}(X)$ is arbitrarily close to $x$. In particular, we show the density of Chern slopes in the pathological Bogomolov–Miyaoka–Yau interval $(3,\infty)$ for any given $p$. Moreover, we prove that for $C=\mathbb{P}^{1}$ there exist surfaces $X$ as above with $H^{1}(X,\mathcal{O}_{X})=0$, that is, with Picard scheme equal to a reduced point. In this way, we show that even surfaces with reduced Picard scheme are densely persistent in $[2,\infty)$ for any given $p$.

#### Article information

Source
Duke Math. J., Volume 166, Number 5 (2017), 975-1004.

Dates
Received: 23 September 2015
Revised: 2 July 2016
First available in Project Euclid: 15 December 2016

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1481771254

Digital Object Identifier
doi:10.1215/00127094-3792596

Mathematical Reviews number (MathSciNet)
MR3626568

Zentralblatt MATH identifier
06707167

#### Citation

Urzúa, Giancarlo. Chern slopes of surfaces of general type in positive characteristic. Duke Math. J. 166 (2017), no. 5, 975--1004. doi:10.1215/00127094-3792596. https://projecteuclid.org/euclid.dmj/1481771254

#### References

• [1] W. P. Barth, K. Hulek, C. A. M. Peters, and A. Van de Ven, Compact Complex Surfaces, Ergeb. Math. Grenzgeb. (3) 4, Springer, Berlin, 2004.
• [2] G. Barthel, F. Hirzebruch, and T. Höfer, Geradenkonfigurationen und Algebraische Flächen, Aspects Math. D4, Vieweg, Braunschweig, 1987.
• [3] I. Bouw, The p-rank of ramified covers of curves, Compos. Math. 126 (2001), 295–322.
• [4] R. Easton, Surfaces violating Bogomolov-Miyaoka-Yau in positive characteristic, Proc. Amer. Math. Soc. 136 (2008), 2271–2278.
• [5] H. Esnault and E. Viehweg, Lectures on Vanishing Theorems, DMV Seminar 20, Birkhäuser, Basel, 1992.
• [6] K. Girstmair, Zones of large and small values for Dedekind sums, Acta Arith. 109 (2003), 299–308.
• [7] K. Girstmair, Continued fractions and Dedekind sums: Three-term relations and distribution, J. Number Theory 119 (2006), 66–85.
• [8] A. Grothendieck, Revêtements étales et groupe fondamental, Séminaire de Géométrie Algébrique du Bois Marie 1960–1961 (SGA 1), Lecture Notes in Math. 224, Springer, Berlin, 1971.
• [9] R. Hartshorne, Algebraic Geometry, Grad. Texts in Math. 52, Springer, New York, 1977.
• [10] F. Hirzebruch and D. Zagier, The Atiyah-Singer Theorem and Elementary Number Theory, Publish or Perish, Boston, 1974.
• [11] J. Jang, Generically ordinary fibrations and a counterexample to Parshin’s conjecture, Michigan Math. J. 59 (2010), 169–178.
• [12] K. Joshi, Crystalline aspects of geography of low dimensional varieties: Numerology, preprint, arXiv:1403.6402v1 [math.AG].
• [13] J. Kollár, Rational Curves on Algebraic Varieties, Ergeb. Math. Grenzgeb. (3) 32, Springer, Berlin, 1996.
• [14] J. Kollár and S. Mori, Birational Geometry of Algebraic Varieties, Cambridge Tracts in Math. 134, Cambridge Univ. Press, Cambridge, 1998.
• [15] A. Langer, Bogomolov’s inequality for Higgs sheaves in positive characteristic, Invent. Math. 199 (2015), 889–920.
• [16] C. Liedtke, “Algebraic surfaces in positive characteristic” in Birational Geometry, Rational Curves, and Arithmetic, Springer, New York, 2013, 229–292.
• [17] K. Mitsui, Homotopy exact sequences and orbifolds, Algebra Number Theory 9 (2015), 1089–1136.
• [18] D. Mumford, Pathologies, III, Amer. J. Math. 89 (1967), 94–104.
• [19] D. Mumford and T. Oda, Algebraic Geometry, II, preprint, www.dam.brown.edu/people/mumford/alg_geom/papers/AGII.pdf (accessed 9 November 2016).
• [20] M. Mustaţă, Toric resolution of singularities, lecture notes, 2004, www.math.lsa.umich.edu/~mmustata/toric_var.html.
• [21] J. P. Murre, Lectures on an Introduction to Grothendieck’s Theory of the Fundamental Group, Tata Inst. Fundam. Res. Stud. Math. 40, Tata Inst. Fundam. Res., Bombay, 1967.
• [22] M. V. Nori, Zariski’s conjecture and related problems, Ann. Sci. Éc. Norm. Supér. (4) 16 (1983), 305–344.
• [23] A. N. Parshin, “The Bogomolov-Miyaoka-Yau inequality for the arithmetical surfaces and its applications” in Séminaire de théorie des nombres (Paris, 1986–1987), Progr. Math. 75, Birkhäuser, Boston, 1988, 299–312.
• [24] A. N. Parshin, “Letter to Don Zagier by A. N. Parshin” in Arithmetic Algebraic Geometry (Texel, 1989), Progr. Math. 89, Birkhäuser, Boston, 1991, 285–292.
• [25] J. Rana, Boundary divisors in the moduli space of stable quintic surfaces, preprint, arXiv:1407.7148v1 [math.AG].
• [26] M. Raynaud, Mauvaise réduction des courbes et $p$-rang, C. R. Math. Acad. Sci. Paris 319 (1994), 1279–1282.
• [27] X. Roulleau and G. Urzúa, Chern slopes of simply connected complex surfaces of general type are dense in [$2,3$], Ann. of Math. (2) 182 (2015), 287–306.
• [28] J.-P. Serre, “Sur la topologie des variétés algébriques en caractéristique $p$” in Symposium internacional de topología, Universidad Nacional Autónoma de México, Mexico City, 1958, 24–53.
• [29] J.-P. Serre, “Espaces fibrés algébriques” in Séminaire Bourbaki, Vol. 2, (1951–1954), no. 82, Soc. Math. France, Paris, 1995, 305–311.
• [30] N. I. Shepherd-Barron, Geography for surfaces of general type in positive characteristic, Invent. Math. 106 (1991), 263–274.
• [31] T. Szamuely, Galois Groups and Fundamental Groups, Cambridge Stud. Adv. Math. 117, Cambridge Univ. Press, Cambridge, 2009.
• [32] L. Szpiro, “Sur le théorème de rigidité de Parshin et Arakelov” in Journées de Géométrie Algébrique de Rennes, II (Rennes, 1978), Astérisque 64, Soc. Math. France, Paris, 1979, 169–202.
• [33] G. Urzúa, Arrangements of curves and algebraic surfaces, Ph.D. dissertation, University of Michigan, Ann Arbor, Mich., 2008, https://deepblue.lib.umich.edu/handle/2027.42/60657.
• [34] G. Urzúa, Arrangements of curves and algebraic surfaces, J. of Algebraic Geom. 19 (2010), 335–365.
• [35] G. Urzúa, Arrangements of rational sections over curves and the varieties they define, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 4 (2011), 453–486.
• [36] G. Xiao, $\pi_{1}$ of elliptic and hyperelliptic surfaces, Internat. J. Math. 2 (1991), 599–615.