Degenerations of K3 surfaces in projective space

Authors:
Francisco Javier Gallego and B. P. Purnaprajna

Journal:
Trans. Amer. Math. Soc. **349** (1997), 2477-2492

MSC (1991):
Primary 14J10, 14J25, 14J28; Secondary 14C05, 14C34, 32G20

MathSciNet review:
1401520

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Abstract | References | Similar Articles | Additional Information

Abstract: The purpose of this article is to study a certain kind of numerical K3 surfaces, the so-called K3 carpets. These are double structures on rational normal scrolls with trivial dualizing sheaf and irregularity . As is deduced from our study, K3 carpets can be obtained as degenerations of smooth K3 surfaces. We also study the Hilbert scheme near the locus parametrizing K3 carpets, characterizing those K3 carpets whose corresponding Hilbert point is smooth. Contrary to the case of canonical ribbons, not all K3 carpets are smooth points of the Hilbert scheme.

**[ACGH]**E. Arbarello, M. Cornalba, P. A. Griffiths, and J. Harris,*Geometry of algebraic curves. Vol. I*, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 267, Springer-Verlag, New York, 1985. MR**770932****[A]**P. Azcue,*On the dimension of the Chow varieties*, Harvard Thesis (1992).**[BE]**Dave Bayer and David Eisenbud,*Ribbons and their canonical embeddings*, Trans. Amer. Math. Soc.**347**(1995), no. 3, 719–756. MR**1273472**, 10.1090/S0002-9947-1995-1273472-3**[CLM]**Ciro Ciliberto, Angelo Lopez, and Rick Miranda,*Projective degenerations of 𝐾3 surfaces, Gaussian maps, and Fano threefolds*, Invent. Math.**114**(1993), no. 3, 641–667. MR**1244915**, 10.1007/BF01232682**[D]**I. V. Dolgachev,*On special algebraic K3 surfaces.*I, Math. USSR Izvestija**7**(1973), 833-846.**[E]**David Eisenbud,*Green’s conjecture: an orientation for algebraists*, Free resolutions in commutative algebra and algebraic geometry (Sundance, UT, 1990) Res. Notes Math., vol. 2, Jones and Bartlett, Boston, MA, 1992, pp. 51–78. MR**1165318****[EG]**David Eisenbud and Mark Green,*Clifford indices of ribbons*, Trans. Amer. Math. Soc.**347**(1995), no. 3, 757–765. MR**1273474**, 10.1090/S0002-9947-1995-1273474-7**[F]**Lung-Ying Fong,*Rational ribbons and deformation of hyperelliptic curves*, J. Algebraic Geom.**2**(1993), no. 2, 295–307. MR**1203687****[GP]**M. Gross & S. Popescu,*Equations of polarized abelian surfaces*, preprint.**[Ha]**Joe Harris,*A bound on the geometric genus of projective varieties*, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4)**8**(1981), no. 1, 35–68. MR**616900****[H]**Robin Hartshorne,*Algebraic geometry*, Springer-Verlag, New York-Heidelberg, 1977. Graduate Texts in Mathematics, No. 52. MR**0463157****[HV]**K. Hulek and A. Van de Ven,*The Horrocks-Mumford bundle and the Ferrand construction*, Manuscripta Math.**50**(1985), 313–335. MR**784147**, 10.1007/BF01168835**[R]**Miles Reid,*Hyperelliptic linear systems on a K3 surface*, J. London Math. Soc. (2)**13**(1976), no. 3, 427–437. MR**0435082****[SP]***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). MR**785216****[S]**Edoardo Sernesi,*Topics on families of projective schemes*, Queen’s Papers in Pure and Applied Mathematics, vol. 73, Queen’s University, Kingston, ON, 1986. MR**869062**

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Additional Information

**Francisco Javier Gallego**

Affiliation:
Departamento de Álgebra, Facultad de Matemáticas, Universidad Complutense de Madrid, 28040 Madrid, Spain

Email:
gallego@eucmos.sim.ucm.es

**B. P. Purnaprajna**

Affiliation:
Department of Mathematics, Brandeis University, Waltham, Massachusetts 02254-9110

Address at time of publication:
Department of Mathematics, Oklahoma State University, Stillwater, Oklahoma 74078

Email:
purna@littlewood.math.okstate.edu

DOI:
http://dx.doi.org/10.1090/S0002-9947-97-01816-3

Received by editor(s):
January 11, 1996

Additional Notes:
We are very pleased to thank our advisor David Eisenbud for his help, patience and encouragement. We would also like to thank Andrea Bruno and Enrique Arrondo for helpful discussions and Mohan Kumar for his helpful comments and suggestions.

Article copyright:
© Copyright 1997
American Mathematical Society