Examples of factorial threefolds complete intersection in $${\mathbb{P}^5}$$ with many singularities

Abstract

Denote by νm(d) the maximal integer for which there exists for \({d \gg 0}\) a threefold \({X\subset \mathbb{P}^5}\) complete intersection of hypersurfaces of degree respectively d and d − 1 such that X has only ordinary singularities of order m and |Sing(X)| = νm(d). We prove that, \({\nu_m(d)\ge \varphi(d)}\) where \({\varphi(d)\sim d^5}\) asymptotically. This result extends (Di Gennaro and Franco in Commun Contemp Math 10(5):745–764, 2008, Corollary 2.10).