Abstract
Motivated by our study (elsewhere) of linear syzygies of homogenous ideals generated by quadrics and their intersections to subvarieties of the ambient projective space, we investigate in this note possible zero-dimensional intersections of two Veronese surfaces in P5.
The case of two Veronese surfaces in P5 meeting in 10 simple points appears also in work of Coble, Conner and Reye in relation to the 10 nodes of a quartic symmetroid in P3, and we provide here a modern account for some of their results.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
M. Beltrametti, A.J. Sommese. Zero cycles and k-th order embeddings of smooth pro- jective surfaces. In: Problems in the theory of surfaces and their classification. Symposia Mathematica XXXII, 33–48 (1991).
A. Cayley, A Memoir on Quartic Surfaces. Proc. London Math. Soc. III, 19–69. Or Collected Works, Tome VII, Math. Ann. 1894.
A.B. Coble, Associated sets of points. Trans Amer. Math. Soc. 24, (1922), 1–20.
J. R. Conner, Basic systems of rational norm-curves. Amer. J. Math. 32 (1911), 115–176.
F. Cossec, Reye congruences. Trans. Amer. Math. Soc. 280 (1983), no. 2, 737–751.
6. D. Eisenbud, M. Green, K. Hulek, S. Popescu, W. Oxbury, Restricting syzygies to linear subspaces. in preparation.
D. Eisenbud, J-H. Koh, M. Stillman. Determinantal equations for curves of high degree. Amer. J. of Math., 110 (1988), 513–539.
D Eisenbud, S. Popescu. The projective geometry of the Gale transformation. J. Algebra 230 (2000), 127–173.
D. Grayson, M. Stillman, Macaulay2: A computer program designed to support computations in algebraic geometry and computer algebra. Source, object code, manual and tutorials are available from http: //www.math.uiuc.edu/Macaulay2/.
Ph. Griffiths, J. Harris. it Principles of algebraic geometry. Wiley-Interscience, New York, 1978.
L. Gruson, Ch. Peskine, Courbes de Vespace projectif: variétés de sécantes. In: Enumerative geometry and classical algebraic geometry (Nice, 1981), Progr. Math., 24, Birkhäuser (1982), 1–31.
R. Hernandez and I. Sols. Line congruences of low degree. Geomètrie algèbrique et applications, II (La Rabida, 1984), 141–154, Travaux en Cours, 23, Hermann, Paris, 1987.
CM. Jessop. A treatise on the line complex. Chelsea Publishing Co., New York 1969.
C.M. Jessop. Quartic surfaces with singular points. Cambridge University Press, 1916.
E.E. Kummer. Abh. Akad. Wiss. Berlin 1866, 1–120.
S. Mukai. Biregular classification of Fano 3-folds and Fano manifolds of coindex 3. Proc. Nat. Acad. Sci. U.S.A. 86 (1989), no. 9, 3000–3002.
S. Mukai, Fano 3-folds. In: Complex projective geometry Trieste, 1989/Bergen, 1989, London Math. Soc. LNS 179, 255–263 (1992).
T. Nakashima. On Reider’s method for surfaces in positive characteristic. J. Reine Angew. Math. 438 (1993), 175–185.
T. Reye. Über lineare Systeme und Gewebe von Fldchen zweiten Grades. J. Reine Angew. Math. 82 (1887), 75–94.
F.-O. Schreyer. Geometry and algebra of prime Fano 3-folds of genus 12 Compositio Math. 127 (2001), no. 3,297–319.
N. Shepherd-Barron. Unstable vector bundles and linear systems on surfaces in characteristic p. Invent. Math. 106 (1991), no. 2, 243–262.
H. Terakawa. The dvery ampleness on a projective surface in positive characteristic. Pacific J. of Math. 187 (1999), no. 1, 187–198.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2003 Springer Science+Business Media Dordrecht
About this paper
Cite this paper
Eisenbud, D., Hulek, K., Popescu, S. (2003). A Note on the Intersection of Veronese Surfaces. In: Herzog, J., Vuletescu, V. (eds) Commutative Algebra, Singularities and Computer Algebra. NATO Science Series, vol 115. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-1092-4_7
Download citation
DOI: https://doi.org/10.1007/978-94-007-1092-4_7
Publisher Name: Springer, Dordrecht
Print ISBN: 978-1-4020-1487-1
Online ISBN: 978-94-007-1092-4
eBook Packages: Springer Book Archive