# The geometric and numerical properties of duality in projective algebraic geometry

- Received:

DOI: 10.1007/BF01259325

- Cite this article as:
- Holme, A. Manuscripta Math (1988) 61: 145. doi:10.1007/BF01259325

## Abstract

In this paper we investigate some fundamental geometric and numerical properties of*duality for projective varieties* in**P**_{k}^{N}=**P**^{N}. We take a point of view which in our opinion is somewhat more*geometric* and less*algebraic* and*numerical* than what has been customary in the literature, and find that this can some times yield simpler and more natural proofs, as well as yield additional insight into the situation. We first recall the standard definitions of the*dual variety* and the*conormal scheme*, introducing classical numerical invariants associated with duality. In section 2 we recall the well known duality properties these invariants have, and which was noted first by*T. Urabe*. In section 3 we investigate the connection between these invariants and*Chern classes* in the singular case. In section 4 we give a treatment of the dual variety of a hyperplane section of X, and the dual procedure of taking the dual of a projection of X. This simplifies the proofs of some very interesting theorems due to*R. Piene*. Section 5 contains a new and simpler proof of a theorem of*A. Hefez and S. L. Kleiman*. Section 6 contains some further results, geometric in nature.