ALGEBRAIC PROPERTIES OF GENERIC SINGULARITIES

Let X (subset) P^n be a nonsingular projective variety of dimension r over an algebraically closed field k which is appropriately embedded if char(k) ≠ 0. By a result of Joel Roberts, there exists a projection π : X (longrightarrow) Pm from a linear center onto X′ = π(X) where r + 1 ≤ m ≤ 2r, such that most of the singularities of X′ are of specific parametric form, and these projections are generic. Following some known results, we give the local defining ideals of these singularities at points where X′ is analytically irreducible. Under a convenient specialization, the local defining ideal of X′ at any such point turns out to be a square-free monomial ideal. We revisit some algebraic properties of the associated simplicial complexes. We also give a depth formula at such points which leads to a partial affirmative answer to a conjecture of Andreotti-Bombieri-Holm on the weak normality of X′.