Abstract
Many recent constructions of varieties, including the lists of K3 surfaces in Magma, use graded ring methods.We show how to apply the method using Magma and, as an application, construct 27 families of K3 surfaces that appear as degenerate cases of surfaces in the usual lists. These are displayed in Tables 1–3 and include both standard degenerations and new examples.
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References
1. Selma Altýnok, Gavin Brown, Miles Reid, Fano 3-folds, K3 surfaces and graded rings, pp. 25–53 in: Topology and geometry: commemorating SISTAG, Contemp. Math. 314, Providence (RI): Amer. Math. Soc., 2002.
2. Selma Altýnok, Hilbert series and applications to graded rings, Int. J. Math. Math. Sci. 7 (2003), 397–403.
3. Wieb Bosma, John Cannon, Catherine Playoust, The Magma algebra system I: The user language, J. Symbolic Comput. 24 (1997), 235–265. See also the Magma home page at http://magma.maths.usyd.edu.au/magma/.
4. Gavin Brown, A database of polarized K3 surfaces, to appear in Exp. Math. (2006). See also www.kent.ac.uk/ims/grdb/
5. Gavin Brown, Datagraphs in algebraic geometry, in: F. Winkler, U. Langer (eds.), Symbolic and Numerical Scientific Computing – Proc. of SNSC'01, LNCS 2630, Heidelberg: Springer-Verlag, 2003.
6. Anita Buckley, Balázs Szendr½oi, Orbifold Riemann-Roch for threefolds with an application to Calabi-Yau geometry, J. Algebraic Geom. 14-4 (2005), 601–622.
7. Anita Buckley, Orbifold Riemann-Roch for 3-folds and applications to Calabi-Yaus, PhD thesis, University of Warwick, 2003, viii+119pp.
8. Igor Dolgachev, Weighted projective varieties, pp. 34–71 in: Group actions and vector fields (Vancouver, B.C., 1981), Lecture Notes in Math. 956, Berlin: Springer, 1982.
9. Robin Hartshorne, Algebraic geometry, Graduate Texts in Mathematics 52, New York: Springer-Verlag, 1977.
10. A. R. Iano-Fletcher, Working with weighted complete intersections pp. 101–173 in: Explicit birational geometry of 3-folds, London Math. Soc. Lecture Note Ser. 281, Cambridge: Cambridge Univ. Press, 2000.
11. Jennifer M. Johnson, János Kollár, Fano hypersurfaces in weighted projective 4-spaces, Experiment. Math. 10-1 (2001), 151–158.
12. Andrew R. Kustin, Matthew Miller, Constructing big Gorenstein ideals from small ones, J. Algebra 85-2 (1983), 303–322.
13. János Kollár, Rational curves on algebraic varieties, Ergebnisse der Mathematik und ihrer Grenzgebiete (3. Folge) 32, Berlin: Springer-Verlag, 1996.
14. Stavros Papadakis, Kustin-Miller unprojection with complexes, J. Algebraic Geom. 13-2 (2004), 249–268.
15. Stavros Papadakis, Miles Reid, Kustin–Miller unprojection with complexes, J. Algebraic Geom. 13-3 (2004), 563–577.
16. Miles Reid, Graded rings and varieties in weighted projective space. Links from www.maths.warwick.ac.uk/miles/surf/
17. Miles Reid, Surfaces with pg = 0, K2 = 1, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 25-1 (1978), 75–92.
18. Miles Reid, Young person's guide to canonical singularities, pp. 345–414 in: Algebraic geometry, Bowdoin, 1985 (Brunswick, Maine, 1985), Proc. Sympos. Pure Math. 46, Providence (RI): Amer. Math. Soc., 1987.
19. Miles Reid, Chapters on algebraic surfaces, pp. 3–159 in: Complex algebraic geometry (Park City, UT, 1993), IAS/Park City Math. Ser. 3, Providence (RI): Amer. Math. Soc., 1997.
20. Miles Reid, Kaori Suzuki, Cascades of projections from log del Pezzo surfaces, pp. 227–249 in: Number theory and algebraic geometry – to Peter Swinnerton- Dyer on his 75th birthday, Cambridge: Cambridge University Press, 2003.
21. B. Saint-Donat, Projective models of K3 surfaces, Amer. J. Math. 96 (1974), 602–639.
22. Kaori Suzuki, On Fano indices of Q-Fano 3-folds, Manuscripta Math. 114-2 (2004), 229–246.
23. Balázs Szendr½oi, Calabi-Yau threefolds with a curve of singularities and counterexamples to the Torelli problem, Internat. J. Math. 11-3 (2000), 449–459.
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Brown, G. (2006). Graded rings and special K3 surfaces. In: Bosma, W., Cannon, J. (eds) Discovering Mathematics with Magma. Algorithms and Computation in Mathematics, vol 19. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-37634-7_6
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DOI: https://doi.org/10.1007/978-3-540-37634-7_6
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