A finite group acting on the moduli space of K3 surfaces

Stellari, Paolo

doi:10.1090/S0002-9947-08-04512-1

A finite group acting on the moduli space of K3 surfaces
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Trans. Amer. Math. Soc. 360 (2008), 6631-6642 Request permission

Abstract:

We consider the natural action of a finite group on the moduli space of polarized K3 surfaces which induces a duality defined by Mukai for surfaces of this type. We show that the group permutes polarized Fourier-Mukai partners of polarized K3 surfaces and we study the divisors in the fixed loci of the elements of this finite group.

References

Géométrie des surfaces $K3$: modules et périodes, Société Mathématique de France, Paris, 1985 (French). Papers from the seminar held in Palaiseau, October 1981–January 1982; Astérisque No. 126 (1985) (1985). MR 785216
Wolf P. Barth, Klaus Hulek, Chris A. M. Peters, and Antonius Van de Ven, Compact complex surfaces, 2nd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 4, Springer-Verlag, Berlin, 2004. MR 2030225, DOI 10.1007/978-3-642-57739-0
Ciprian Borcea, $K3$ surfaces and complex multiplication, Rev. Roumaine Math. Pures Appl. 31 (1986), no. 6, 499–505. MR 856201
Tom Bridgeland and Antony Maciocia, Complex surfaces with equivalent derived categories, Math. Z. 236 (2001), no. 4, 677–697. MR 1827500, DOI 10.1007/PL00004847
I. V. Dolgachev, Mirror symmetry for lattice polarized $K3$ surfaces, J. Math. Sci. 81 (1996), no. 3, 2599–2630. Algebraic geometry, 4. MR 1420220, DOI 10.1007/BF02362332
Brendan Hassett, Special cubic fourfolds, Compositio Math. 120 (2000), no. 1, 1–23. MR 1738215, DOI 10.1023/A:1001706324425
Shinobu Hosono, Bong H. Lian, Keiji Oguiso, and Shing-Tung Yau, Fourier-Mukai number of a K3 surface, Algebraic structures and moduli spaces, CRM Proc. Lecture Notes, vol. 38, Amer. Math. Soc., Providence, RI, 2004, pp. 177–192. MR 2096145, DOI 10.1090/crmp/038/08
D. R. Morrison, On $K3$ surfaces with large Picard number, Invent. Math. 75 (1984), no. 1, 105–121. MR 728142, DOI 10.1007/BF01403093
Shigeru Mukai, Duality between $D(X)$ and $D(\hat X)$ with its application to Picard sheaves, Nagoya Math. J. 81 (1981), 153–175. MR 607081
S. Mukai, On the moduli space of bundles on $K3$ surfaces. I, Vector bundles on algebraic varieties (Bombay, 1984) Tata Inst. Fund. Res. Stud. Math., vol. 11, Tata Inst. Fund. Res., Bombay, 1987, pp. 341–413. MR 893604
Shigeru Mukai, Duality of polarized $K3$ surfaces, New trends in algebraic geometry (Warwick, 1996) London Math. Soc. Lecture Note Ser., vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 311–326. MR 1714828, DOI 10.1017/CBO9780511721540.012
V.V. Nikulin, Integral symmetric bilinear forms and some of their applications, Math. USSR Izvestija 14 (1980), 103–167.
V.V. Nikulin, Correspondences of a K3 surfaces with itself via moduli of sheaves. I, math.AG/0609233.
Keiji Oguiso, K3 surfaces via almost-primes, Math. Res. Lett. 9 (2002), no. 1, 47–63. MR 1892313, DOI 10.4310/MRL.2002.v9.n1.a4
D. O. Orlov, Equivalences of derived categories and $K3$ surfaces, J. Math. Sci. (New York) 84 (1997), no. 5, 1361–1381. Algebraic geometry, 7. MR 1465519, DOI 10.1007/BF02399195
B. Saint-Donat, Projective models of $K-3$ surfaces, Amer. J. Math. 96 (1974), 602–639. MR 364263, DOI 10.2307/2373709
Francesco Scattone, On the compactification of moduli spaces for algebraic $K3$ surfaces, Mem. Amer. Math. Soc. 70 (1987), no. 374, x+86. MR 912636, DOI 10.1090/memo/0374
Paolo Stellari, Some remarks about the FM-partners of $K3$ surfaces with Picard numbers 1 and 2, Geom. Dedicata 108 (2004), 1–13. MR 2112663, DOI 10.1007/s10711-004-9291-7
Hans Sterk, Finiteness results for algebraic $K3$ surfaces, Math. Z. 189 (1985), no. 4, 507–513. MR 786280, DOI 10.1007/BF01168156
Joachim Wehler, $K3$-surfaces with Picard number $2$, Arch. Math. (Basel) 50 (1988), no. 1, 73–82. MR 925498, DOI 10.1007/BF01313498