Book contents
- Frontmatter
- Contents
- INTRODUCTION
- PROJECTIVE VARIETIES
- VECTOR BUNDLES AND SPECIAL PROJECTIVE EMBEDDINGS
- LIST OF PARTICIPANTS
- Speciality one rational surfaces in P4
- Bounding sections of bundles on curves
- The smooth surfaces of degree 9 in P4
- Compactifying the space of elliptic quartic curves
- Threefolds of degree 11 in P5
- Complete extensions and their map to moduli space
- On the Betti numbers of the moduli space of stable bundles of rank two on a curve
- Gaussian maps for certain families of canonical curves
- Geometry of the Horrocks bundle on P3
- Stability and restrictions of Picard bundles, with an application to the normal bundles of elliptic curves
- Sections planes et majoration du genre des courbes gauches
- A tribute to Corrado Segre
- Un aperçu des travaux mathématiques de G.H. Halphen (1844–1889)
- The source double-point cycle of a finite map of codimension one
- Fibré déterminant et courbes de saut sur les surfaces algébriques
- Courbes minimales dans les classes de biliaison
- Fano 3-folds
- Polarized K3 surfaces of genus 18 and 20
- Protective compactifications of complex afflne varieties
- On generalized Laudal's lemma
- Sur la stabilité des sous-variétés lagrangiennes des variétés symplectiques holomorphes
- Introduction to Gaussian maps on an algebraic curve
- Some examples of obstructed curves in P3
The smooth surfaces of degree 9 in P4
Published online by Cambridge University Press: 06 July 2010
- Frontmatter
- Contents
- INTRODUCTION
- PROJECTIVE VARIETIES
- VECTOR BUNDLES AND SPECIAL PROJECTIVE EMBEDDINGS
- LIST OF PARTICIPANTS
- Speciality one rational surfaces in P4
- Bounding sections of bundles on curves
- The smooth surfaces of degree 9 in P4
- Compactifying the space of elliptic quartic curves
- Threefolds of degree 11 in P5
- Complete extensions and their map to moduli space
- On the Betti numbers of the moduli space of stable bundles of rank two on a curve
- Gaussian maps for certain families of canonical curves
- Geometry of the Horrocks bundle on P3
- Stability and restrictions of Picard bundles, with an application to the normal bundles of elliptic curves
- Sections planes et majoration du genre des courbes gauches
- A tribute to Corrado Segre
- Un aperçu des travaux mathématiques de G.H. Halphen (1844–1889)
- The source double-point cycle of a finite map of codimension one
- Fibré déterminant et courbes de saut sur les surfaces algébriques
- Courbes minimales dans les classes de biliaison
- Fano 3-folds
- Polarized K3 surfaces of genus 18 and 20
- Protective compactifications of complex afflne varieties
- On generalized Laudal's lemma
- Sur la stabilité des sous-variétés lagrangiennes des variétés symplectiques holomorphes
- Introduction to Gaussian maps on an algebraic curve
- Some examples of obstructed curves in P3
Summary
Introduction
The classification of smooth surfaces of low degree d in P4 goes back to the Italian geometers at the turn of the century. They treated the cases d ≤ 6. More recently, Ionescu and Okonek have treated the cases d = 7 and d = 8 ([Io],[O1, O2]). Their work were complemented by a result of Alexander to give a classification ([A1]). In this paper we find the possible numerical invariants of surfaces of degree 9, and describe for each set of invariants the family of surfaces with the given invariants. Some of the results are mentioned in [Ra], which deals with the case d = 10.
We work over an algebraically closed field of characteristic 0.
The first result is the following
Theorem. Let S be a smooth nondegenerate surface of degree 9 in P4 with sectional genus π, Euler-Poincaré characteristic χ and canonical class K. Then S is a regular surface with K2 = 6χ – 5π + 23, where
π = 6 and χ = 1 and S is rational or S is the projection of an Enriques surface of degree 10 in P5 with center of projection on the surface, or
π = 7 and χ = 1 and S is a rational surface, or χ = 2 and S is a minimal properly elliptic surface, or
π = 8 and χ = 2 and S is a K3-surface with five (−1)-lines, or χ = 3 and S is a minimal surface of general type, or
π = 9 and χ = 4 and S is linked (3,4) to a cubic scroll (possibly singular/reducible), or
π = 10 and χ = 5 and S is a complete intersection (3,3), or
π = 12 and χ = 9 and S is linked (2,5) to a plane.
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- Complex Projective GeometrySelected Papers, pp. 32 - 46Publisher: Cambridge University PressPrint publication year: 1992
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