Maximal families of nodal varieties with defect

Abstract

AbstractIn this paper we prove that a nodal hypersurface in $$\mathbf {P}^4$$ P 4 with positive defect has at least $$(d-1)^2$$ ( d - 1 ) 2 nodes, and if it has at most $$2(d-2)(d-1)$$ 2 ( d - 2 ) ( d - 1 ) nodes and $$d\ge 7$$ d ≥ 7 then it contains either a plane or a quadric surface. Furthermore, we prove that a nodal double cover of $$\mathbf {P}^3$$ P 3 ramified along a surface of degree 2d with positive defect has at least $$d(2d-1)$$ d ( 2 d - 1 ) nodes. We construct the largest dimensional family of nodal degree d hypersurfaces in $$\mathbf {P}^{2n+2}$$ P 2 n + 2 with positive defect for d sufficiently large.