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\(K3\) surfaces with 9 cusps in characteristic \(p\). (English) Zbl 1448.14035

The complex \(K3\) surfaces with \(9\) cusps have been studied by topological methods in [W. Barth, Geom. Dedicata 72, No. 2, 171–178 (1998; Zbl 1035.14014); in: Complex analysis and algebraic geometry. A volume in memory of Michael Schneider. Berlin: Walter de Gruyter. 41–59 (2000; Zbl 1067.14509)]. The aim of the current paper is to investigate the same problem on any algebraically closed field \(k\) of characteristic \(p \neq 3\). The main results of the paper, including Theorems 1.1 and 1.2, can be restated as follows.
Theorem 1. Let \(X\) be a \(K3\) surface with \(9\) cusps defined over \(k\) of characteristic \(\neq 3\). Then \(X\) admits a triple covering by an abelian surface with an automorphism of order 3. Moreover, if \(X\) is supersingular, then either \(X\) has Artin invariant \(\sigma=1\), or \(X\) has Artin invariant \(\sigma=2\) and \(p\equiv -1\bmod 3\)
In order to support their result, they provide ample examples containing abelian surfaces with an automorphism of order \(3\), which are treated in Proposition 1.3 and Theorem 1.4, as follows:
Theorem 2. Let \(A\) be an abelian surface with an automorphism \(\sigma\) of order 3.
1) If \(A \sim E_1 \times E_2\), with non-isogenous elliptic curves \(E_i\), then the quotient \(A/\left<\sigma\right>\) is not birationally equivalent to a \(K3\) surface.
2) If \(A\) is a simple ordinary abelian variety, and \(\sigma\) is not a translation, then \(A/\left<\sigma\right>\) is birationally equivalent to a \(K3\) surface.
In Section 2, the authors included the Lattice theory of \(K3\) surfaces with 9 cusps over algebraically fields of characteristic \(\neq 3\). The proof of above Theorem 1 is given in Sections 3 and 4, and Theorem 2 is proved through the sections 5 to 7.
Finally, in Section 8, they exhibited explicit families of \(K3\) surfaces with nine cusps in any characteristic \(\neq 3\) such that covering abelian surfaces are generically simple.

MSC:

14J28 \(K3\) surfaces and Enriques surfaces
11G10 Abelian varieties of dimension \(> 1\)
14J27 Elliptic surfaces, elliptic or Calabi-Yau fibrations
14K05 Algebraic theory of abelian varieties
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References:

[1] Artin, M., Supersingular K3 surfaces, Ann. Sci. Éc. Norm. Supér., 4, 543-568 (1974) · Zbl 0322.14014
[2] Barth, W., K3 surfaces with nine cusps, Geom. Dedic., 72, 171-178 (1998) · Zbl 1035.14014
[3] Barth, W., On the classification of K3 surfaces with nine cusps, (Peternell, T.; Schreyer, F.-O., Complex Analysis and Algebraic Geometry - A Volume in Memory of Michael Schneider (2000)), 42-59 · Zbl 1067.14509
[4] Bonfanti, M. A.; van Geemen, B., Abelian surfaces with an automorphism and quaternionic multiplication, Can. J. Math., 68, 24-43 (2016) · Zbl 1337.14035
[5] Harder, G.; Narashimhan, M. S., On the cohomology group of moduli spaces of vector bundles on curves, Math. Ann., 212, 215-248 (1975) · Zbl 0324.14006
[6] Katsura, T., Generalized Kummer surfaces and their unirationality in characteristic p, J. Fac. Sci., Univ. Tokyo, Sect. 1A, Math., 34, 1-41 (1987) · Zbl 0664.14023
[7] Katsura, T.; Schütt, M., Zariski K3 surfaces, Rev. Mat. Iberoam., 36, 869-894 (2020) · Zbl 1441.14129
[8] Kondō, S.; Shimada, I., On certain duality of Néron-Severi lattices of supersingular K3 surfaces, Algebr. Geom., 1, 311-333 (2014) · Zbl 1322.14057
[9] Lieblich, M.; Maulik, D., A note on the cone conjecture for K3 surfaces in positive characteristic, Math. Res. Lett., 25, 1879-1891 (2018) · Zbl 1420.14086
[10] Matsusaka, T.; Mumford, D., Two fundamental theorems on deformations of polarized varieties, Am. J. Math., 86, 668-684 (1964) · Zbl 0128.15505
[11] Milne, J. S., Etale Cohomology (1980), Princeton Univ. Press: Princeton Univ. Press Princeton, New Jersey · Zbl 0433.14012
[12] Miranda, R., Triple covers in algebraic geometry, Am. J. Math., 107, 1123-1158 (1985) · Zbl 0611.14011
[13] Mumford, D., Abelian Varieties (1970), Oxford Univ. Press: Oxford Univ. Press London/New York · Zbl 0198.25801
[14] Nikulin, V. V., Integral symmetric bilinear forms and some of their applications, Math. USSR, Izv., 14, 1, 103-167 (1980) · Zbl 0427.10014
[15] A. Ogus, Supersingular K3 crystals, Journées de Géométrie Algébrique de Rennes (Rennes, 1978), vol. II, Astérisque, vol. 64, Société Mathématique de France, Paris, pp. 3-86.
[16] Oort, F., Which abelian varieties are products of elliptic curves?, Math. Ann., 214, 35-47 (1975) · Zbl 0283.14007
[17] Pinkham, H., Singularities rationnelles de surfaces, Lecture Notes in Math., vol. 777, 147-178 (1980), Springer-Verlag: Springer-Verlag Berlin, Heidelberg · Zbl 0459.14009
[18] Schütt, M., \( \mathbb{Q}_\ell \)-cohomology projective planes and Enriques surfaces in characteristic two, Épijournal Géom. Algébr., 3, 24 (2019), pp. · Zbl 1505.14085
[19] Schütt, M., Divisibilities among nodal curves, Math. Res. Lett., 25, 1359-1368 (2018) · Zbl 1407.14031
[20] Schütt, M.; Shioda, T., Elliptic surfaces, algebraic geometry in East Asia - Seoul 2008, Adv. Stud. Pure Math., 60, 51-160 (2010) · Zbl 1216.14036
[21] Shioda, T.; Mitani, N., Singular abelian surfaces and binary quadratic forms, classification of algebraic varieties and compact complex manifolds, Lect. Notes Math., 412, 259-287 (1974) · Zbl 0302.14011
[22] Sterk, H., Finiteness results for algebraic K3 surfaces, Math. Z., 189, 507-513 (1985) · Zbl 0545.14032
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