Smoothings of ropes on curves

In this article we present a unified way to smooth multiple structures over smooth varieties. We are interested in certain kinds of multiple structures of dimension 1 called “ropes”. These are schemes which locally look like Y × k[x1,...,xn] (x1,...,xn)2 . By smoothing a rope we mean that the rope is the flat limit of a family of smooth curves. To illustrate the general theory we will focus on the smoothing of ropes of multiplicity 3 over P1. To construct a smoothing we develop a method that connects, on the one hand, first order infinitesimal deformations of a finite morphism to projective space and, on the other hand, morphisms from a rope to projective space. In our arguments we also use the theory of moduli spaces, Hilbert schemes and quot schemes. Our method yields also the smoothing of so called K3 carpets on rational normal scrolls and Enriques surfaces. This application will appear elsewhere. 1. Ropes of multiplicity 3 and its projective embeddings We start with embeddings supported on a rational normal curve lying in the same projective space where the rope is embedded. The even genus case is the following theorem. Theorem 1.1. Let 0 ≤ n < d and N ≥ d+ n− 1 be integers. Let denote E = OP1(−d+ n)⊕ OP1(−d− n). Let Ỹ be a 3-rope over P with conormal bundle E . (1) Then Ỹ is embedded in P supported on a rational normal curve Y ⊂ P . (2) Let X π −→ P be a smooth connected triple cover such that π∗OX = OP1 ⊕ E . Let denote L = πOP1(N). Then L is nonspecial. Remark 1.2. (1) pa(Ỹ ) = g(X) = 2d− 2. (2) H(E ⊗ OP1(N)) = 0. (3) H(E ⊗ OP1(N)) = 0 iff n = 0 and N = d− 1. So the rope Ỹ is linearly normal embedded just in the case n = 0 and N = d− 1. And now the odd genus case. In the odd case we never have H(E ⊗OP1(N)) = 0 so the rope is nonlinearly normal embedded . Theorem 1.3. Let 0 ≤ n < d and N ≥ d+ n be integers. Let denote E = OP1(−d+ n)⊕ OP1(−d− n− 1). Let Ỹ be a 3-rope over P with conormal bundle E . (1) Then Ỹ is embedded in P supported on a rational normal curve Y ⊂ P . (2) Let X π −→ P be a smooth connected triple cover such that π∗OX = OP1 ⊕ E . Let denote L = πOP1(N). Then L is nonspecial.