We show that the arithmetically Cohen–Macaulay (ACM) curves of degree 4 and genus 0 in P 4 form an irreducible subset of the Hilbert scheme. Using this, we show that the singular locus of the corresponding component of the Hilbert scheme has dimension greater than 6. Moreover, we describe the structures of all ACM curves of Hilb 4m+1 (P 4).
We show that any smooth Q-normal lattice polytope P of dimension n and degree d is a strict Cayley polytope if n ≥ 2d + 1. This gives a sharp answer, for this class of polytopes, to a question raised by
We continue the development of methods for enumerating nodal curves on smooth complex surfaces, stressing the range of validity. We illustrate the new methods in three important examples. First, for up to eight nodes, we confirm Göttsche's conjecture about plane curves of low degree. Second, we justify Vainsencher's enumeration of irreducible six-nodal… (More)