On the Canonical Map of Smooth Surfaces in ℙ⁴

Abstract

We prove that there exists an integer M such that if S ⊂ ℙ⁴ is a smooth surface with deg(S) > M, then the canonical map of S is birational. Then we consider surfaces, S, satisfying h ⁱ (ℐ _S (3 − i)) = 0, 0 ≤ i ≤ 2 and show that they are regular and that their canonical system is base point free and very ample if deg(S) > M and S doesn't contain −2-curves.

Key Words:

• Algebraic geometry
• Surfaces
• Canonical map

1991 Mathematics Subject Classification:

• Primary 14J29, 14E05
• Secondary 14M07, 14J99

Acknowledgment

The first author was supported in part by MUIR project “Geometry on algebraic varieties”.

Notes

^#Communicated by S. Kleiman.