K3 carpets on minimal rational surfaces and their smoothings

Abstract

In this article, we study K3 double structures on minimal rational surfaces Y . The results show there are infinitely many abstract K3 double structures on Y parametrized by P 1 , countably many of which are projective. We show that all projective K3 carpets can be smoothed to a smooth K3 surface. One of the byproducts of the proof shows that unless Y is embedded as a variety of minimal degree, there are infinitely many embedded K3 carpet structures on Y . Moreover, we show any embedded projective K3 carpet on F e with e < 3 arises as a flat limit of embeddings degenerating to 2 : 1 morphism. The rest do not, but we still prove the smoothing result. We further show that the Hilbert points corresponding to the non-split K3 carpets supported on F e are smooth points if and only if 0 ≤ e ≤ 2. In contrast, Hilbert points corresponding to nonsplit K3 carpets supported on P 2 are always smooth. The results in [BGG20] show that there are no higher dimensional analogues of the results in this article. results than those in [GP97] and that it would be interesting to do so. The conversation and results in [BGG20] motivated this article.Convention. We will always work over the complex numbers C. For a smooth variety X , K X denotes the canonical bundle of X . The symbol "∼" stands for linear equivalence and the symbol "≡" stands for numerical equivalence of line bundles or divisors. ABSTRACT AND EMBEDDED K3 CARPETSThis section is to investigate the existence of of K3 carpets supported on a minimal rational surface S. We will deal with two cases, namely when S = F e and when S = P 2 . We start with some basic things on ribbons and carpets.2.1. Ropes, ribbons and K3 carpets. Ropes are multiple structures on a scheme and ribbons are special kinds of ropes. We give the precise definitions below. Definition 2.1. Let Y be a reduced connected scheme and let E be a vector bundle of rank m − 1 on Y. A rope of multiplicity m on Y with conormal bundle E is a scheme Y with Y r ed = Y such that I 2 Y /Y ′ = 0, and I Y /Y ′ = E as O Y modules. If E is a line bundle then Y is called a ribbon on Y .