In this paper, we study K3 double structures on minimal rational surfaces [Formula: see text]. The results show there are infinitely many non-split abstract K3 double structures on [Formula: see text] parametrized by [Formula: see text], countably many of which are projective. For [Formula: see text] there exists a unique non-split abstract K3 double structure which is non-projective (see [J.-M. Drézet, Primitive multiple schemes, preprint (2020), arXiv:2004.04921 , to appear in Eur. J. Math.]). 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 [Formula: see text] is embedded as a variety of minimal degree, there are infinitely many embedded K3 carpet structures on [Formula: see text]. Moreover, we show any embedded projective K3 carpet on [Formula: see text] with [Formula: see text] 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 projective K3 carpets supported on [Formula: see text], embedded by a complete linear series are smooth points if and only if [Formula: see text]. In contrast, Hilbert points corresponding to projective (split) K3 carpets supported on [Formula: see text] and embedded by a complete linear series are always smooth. The results in [P. Bangere, F. J. Gallego and M. González, Deformations of hyperelliptic and generalized hyperelliptic polarized varieties, preprint (2020), arXiv:2005.00342 ] show that there are no higher dimensional analogues of the results in this paper.

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In this article we prove new results on projective normality and normal presentation of adjunction bundle associated to an ample and globally generated line bundle on higher dimensional smooth projective varieties with nef canonical bundle. As one of the consequences of the main theorem, we give bounds on very ampleness and projective normality of pluricanonical linear systems on varieties of general type in dimensions three, four and five. These improve known such results.Ein and Lazarsfeld proved that for a very ample line bundle L on a smooth projective variety X, K X + (n + 1 + p)A satisfies the property N p . (see [10]).Another very interesting and related conjecture is the conjecture by Fujita. The precise statement is the following:Fujita's Conjecture: On a smooth projective variety of dimension n, K X + (n + 1)A is globally generated and K X + (n + 2)A is very ample where A is an arbitrary ample line bundle.Fujita's conjecture has been proved for surfaces by Reider (cf. [37]) using Bogomolov's instability theorem (see [2]) on rank two vector bundles. Fujita's freeness conjecture has been proved by Ein and Lazarsfeld (see [9]) for n = 3, by Kawamata (see [24]) for n = 3, 4 and by Fei Ye and Zhixian Zhu (see [39]) for n = 5.Since the first and the last terms are zero by Kodaira vanishing, hence H i (M L ⊗ (K + (l − i)B)) = 0 for all 2 ≤ i ≤ n.

We show that given an embedding of an Enriques manifold in a large enough projective space there will exist embedded multiple structures with same invariants (cohomology of the structure sheaf) as its universal cover which is either a hyperkähler or a Calabi-Yau manifold. We then show that these multiple structures can be smoothed to smooth hyperkähler or Calabi-Yau manifolds respectively. Hence we obtain a flat family of hyperkähler (or Calabi-Yau) manifolds embedded in the same projective space which degenerates to an embedded multiple structure of the given Enriques manifold. The above shows that the multiple structures on the embedded Enriques manifold are points of the Hilbert scheme containing the fibres of the above family. We prove that if the index of the Enriques manifold is two then they are smooth points of the Hilbert scheme.

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 .