SMOOTH, ISOLATED CURVES IN FAMILIES OF CALABI-YAU THREEFOLDS IN HOMOGENEOUS SPACES

- Journal title : Journal of the Korean Mathematical Society
- Volume 50, Issue 5, 2013, pp.1033-1050
- Publisher : The Korean Mathematical Society
- DOI : 10.4134/JKMS.2013.50.5.1033

Title & Authors

SMOOTH, ISOLATED CURVES IN FAMILIES OF CALABI-YAU THREEFOLDS IN HOMOGENEOUS SPACES

Knutsen, Andreas Leopold;

Knutsen, Andreas Leopold;

Abstract

We show the existence of smooth isolated curves of different degrees and genera in Calabi-Yau threefolds that are complete intersections in homogeneous spaces. Along the way, we classify all degrees and genera of smooth curves on BN general K3 surfaces of genus , where . By results of Mukai, these are the K3 surfaces that can be realised as complete intersections in certain homogeneous spaces.

Keywords

isolated curves;deformations;BN general K3 surfaces;Calabi-Yau threefolds;

Language

English

Cited by

References

1.

M. Arap and N. Marshburn, Brill-Noether general curves on Knutsen K3 surfaces, arXiv:1208.4957.

2.

M. Arap, J. Cutrone, and N. Marshburn, On the existence of certain weak Fano three- folds of Picard number two, arXiv:1112.2611.

3.

W. Barth, K. Hulek, C. Peters, and A. Van de Ven, Compact Complex Surfaces, 2nd edn. Springer-Verlag, Berlin, 2004.

4.

B. H. Gross and N. R. Wallach, On the Hilbert polynomials and Hilbert series of homogeneous projective varieties, Arithmetic geometry and automorphic forms, 253-263, Adv. Lect. Math. (ALM), 19, Int. Press, Somerville, MA, 2011.

5.

J. Harris and I. Morrison, Moduli of Curves, Graduate Texts in Mathematics, 187, Springer-Verlag, New York, 1998.

6.

T. Johnsen and A. L. Knutsen, K3 Projective Models in Scrolls, Lecture Notes in Mathematics, 1842. Springer-Verlag, Berlin, 2004.

7.

H. P. Kley, Rigid curves in complete intersection Calabi-Yau threefolds, Compositio Math. 123 (2000), no. 2 185-208.

9.

A. L. Knutsen, On isolated smooth curves of low genera in Calabi-Yau complete intersection threefolds, Trans. Amer. Math. Soc. 364 (2012), no. 10, 5243-5264.

10.

K. Kodaira, On the structure of compact complex analytic surfaces. I, Amer. J. Math. 86 (1964), 751-798.

11.

R. Lazarsfeld, Brill-Noether-Petri without degenerations, J. Differential Geom. 23 (1986), no. 3, 299-307.

12.

S. Mukai, Curves, K3 surfaces and Fano 3-folds of genus ${\leq}$ 10, in Algebraic geometry and commutative algebra, Vol. I, 357-377, Kinokuniya, Tokyo, 1988.

13.

S. Mukai, New development of theory of Fano 3-folds: vector bundle method and moduli problem, Sugaku 47 (1995), no. 2, 125-144; translation in: Sugaku Expositions 15 (2002), no. 2, 125-150.

15.

X. Yu, On smooth and isolated curves in general complete intersection Calabi-Yau three-folds, arXiv:1208.6282.