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Examples of Calabi–Yau Threefolds as Covers of Almost-Fano Threefolds

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Abstract

We give a method for producing examples of Calabi–Yau threefolds as covers of degree d ≤ 8 of almost-Fano threefolds, computing explicitely their Euler– Poincaré characteristic. Such a method generalizes the well-knownclassical construction of Calabi–Yau threefolds as double covers of the projective space branched along octic surfaces.

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References

  1. Casnati G. (1996). Covers of algebraic varieties II Covers of degree 5 and construction of surfaces. J. Algebraic Geom. 5:461–477

    MATH  MathSciNet  Google Scholar 

  2. Casnati G. (2001). Cover of algebraic varieties IV A Bertini theorem for scandinavian covers of degree 6. Forum Math. 13:21–36

    Article  MATH  MathSciNet  Google Scholar 

  3. Casnati G. (2003). Cover of algebraic varieties V Examples of covers of degree 8 and 9 as catalecticant loci. J. Pure Appl. Algebra 182:17–32

    Article  MATH  MathSciNet  Google Scholar 

  4. Casnati G., Ekedahl T. (1996). Covers of algebraic varieties I A general structure theorem, covers of degree 3, 4 and Enriques surfaces. J. Algebraic Geom. 5:439–460

    MATH  MathSciNet  Google Scholar 

  5. Casnati G., Supino P. (2002). Construction of threefold with finite canonical map. Glasgow Math. J. 44:65–79

    Article  MATH  MathSciNet  Google Scholar 

  6. Catanese F. (1984). On the moduli spaces of surfaces of general type. J. Differential Geom. 19:483–515

    MathSciNet  Google Scholar 

  7. Chang, M.C.: On the Euler numbers of threefolds. In: Yang, L., Yau, S.-T. (eds.) First International Congress of Chinese Mathematicians, vol. 20, pp. 229–234. Beijing, 1998, AMS/IP Stud. Adv. Math., (2001)

  8. Chang M.C., Lopez A.F. (2001). A linear bound on the Euler number of threefolds of Calabi–Yau and of general type. Manuscripta Math. 105:47–67

    Article  MATH  MathSciNet  Google Scholar 

  9. Cynk S. (1999). Double coverings of octic arrangements with isolated singularities. Adv. Theor. Math. Phys. 3:217–225

    MATH  MathSciNet  Google Scholar 

  10. Cynk S. (2003). Cyclic coverings of Fano threefolds. Ann. Polon. Math. 80:117–124

    Article  MATH  MathSciNet  Google Scholar 

  11. Cynk, S., Szemberg, T.: Double covers and Calabi–Yau varieties. In: Jakubczyk, B., Pawłucki, W., Stasica, J. (eds.) Singularities Symposium–Lojasiewicz 70, vol. 44, pp. 93–101. Banach Center Publ., (1998)

  12. Gallego F.J., Purnaprajna B.P. (2003). On the canonical rings of covers of surfaces of minimal degree. Trans. Amer. Math. Soc. 355:2715–2732

    Article  MATH  MathSciNet  Google Scholar 

  13. Goto S., Watanabe K-i (1978). On graded rings I. J. Math. Soc. Japan 30:179–213

    Article  MATH  MathSciNet  Google Scholar 

  14. Harris J., Tu L.W. (1984). On symmetric and skew–symmetric determinantal varieties. Topology 23:71–84

    Article  MATH  MathSciNet  Google Scholar 

  15. Hartshorne, R.: Algebraic geometry. Springer (1977)

  16. Hunt B. (1990). A bound on the Euler number for certain Calabi–Yau 3–folds. J. Reine Angew. Math. 411:137–170

    MATH  MathSciNet  Google Scholar 

  17. Iskovskikh, V.A., Prokhorov, Y.G.: Fano varieties. Algebraic Geometry V (A.N. Parshin and I.R.~Shafarevich, eds.), Encyclopedia of Mathematical Sciences, vol. 47, (1999)

  18. Jahnke P., Peternell T., Radloff I. (2005). Threefolds with big and nef anticanonical bundles I. Math. Ann. 333:569–631

    Article  MathSciNet  MATH  Google Scholar 

  19. Miranda R. (1985). Triple covers in algebraic geometry. Amer. J. Math. 107:1123–1158

    Article  MATH  MathSciNet  Google Scholar 

  20. Pragacz P. (1988). Enumerative geometry of degeneracy loci. Ann. Scient. École. Norm. Sup. 21:413–454

    MATH  MathSciNet  Google Scholar 

  21. Schreyer F.O. (1986). Syzygies of canonical curves and special linear series. Math. Ann. 275:105–137

    Article  MATH  MathSciNet  Google Scholar 

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Casnati, G. Examples of Calabi–Yau Threefolds as Covers of Almost-Fano Threefolds. Geom Dedicata 119, 169–179 (2006). https://doi.org/10.1007/s10711-006-9067-y

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