In this article we study the deformations of hyperelliptic polarized varieties (X, L) of dimension m and sectional genus g such that the image Y of the morphism ϕ induced by |L| is smooth. If L m < 2g − 2, it is known that, by adjunction and the Clifford's theorem, any deformation of (X, L) is hyperelliptic. Thus, we focus on when L m = 2g − 2 or L m = 2g. We prove that, if (X, L) is Fano-K3, then, except when Y is a hyperquadric, all deformations of (X, L) are again hyperelliptic (if Y is a hyperquadric, the general deformation of ϕ is an embedding). This contrasts with the situation of hyperelliptic canonical curves and hyperelliptic K3 surfaces. If L m = 2g, then we prove that, in most cases, a general deformation of ϕ is a finite morphism of degree 1. This provides interesting examples of degree 2 morphisms that can be deformed to morphisms of degree 1. We extend our results to so-called generalized hyperelliptic polarized Fano, Calabi-Yau and general type varieties. The solutions to these questions are closely intertwined with the existence or non existence of double structures on the algebraic varieties Y . We address this matter as well.

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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.

The purpose of this article is twofold. Firstly, we address and completely solve the following question: Let (X, L) be a smooth, hyperelliptic polarized variety and let $$\varphi : X \longrightarrow Y \subset \textbf{P}^N$$ φ : X ⟶ Y ⊂ P N be the morphism induced by |L|; when does $$\varphi $$ φ deform to a birational map? Secondly, we introduce the notion of “generalized hyperelliptic varieties” and carry out a study of their deformations. Regarding the first topic, we settle the non trivial, open cases of (X, L) being Fano-K3 and of (X, L) having dimension $$m \ge 2$$ m ≥ 2 , sectional genus g and $$L^m=2g$$ L m = 2 g . This was not addressed by Fujita in his study of hyperelliptic polarized varieties and requires the introduction of new methods and techniques to handle it. In the Fano-K3 case, all deformations of (X, L) are again hyperelliptic except if Y is a hyperquadric. By contrast, in the $$L^m=2g$$ L m = 2 g case, with one exception, a general deformation of $$\varphi $$ φ is a finite birational morphism. This is especially interesting and unexpected because, in the light of earlier results, $$\varphi $$ φ rarely deforms to a birational morphism when Y is a rational variety, as is our case. The Fano-K3 case contrasts with canonical morphisms of hyperelliptic curves and with hyperelliptic K3 surfaces of genus $$g \ge 3$$ g ≥ 3 . Regarding the second topic, we completely answer the question for generalized hyperelliptic polarized Fano and Calabi–Yau varieties. For generalized hyperelliptic varieties of general type we do this in even greater generality, since our result holds for Y toric. Standard methods in deformation theory do not work in the present setting. Thus, to settle these long standing open questions, we bring in new ideas and techniques building on those introduced by the authors concerning deformations of finite morphisms and the existence and smoothings of certain multiple structures. We also prove a new general result on unobstructedness of morphisms that factor through a double cover and apply it to the case of generalized hyperelliptic varieties.

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 .

Let E be a vector bundle on a smooth projective variety X ⊆ P N that is Ulrich with respect to the hyperplane section H. In this article, we study the Koszul property of E , the slope-semistability of the k-th iterated syzygy bundle S k (E ) for all k ≥ 0 and rationality of moduli spaces of slope-stable bundles on del-Pezzo surfaces. As a consequence of our study, we show that if X is a del-Pezzo surface of degree d ≥ 4, then any Ulrich bundle E satisfies the Koszul property and is slope-semistable. We also show that, for infinitely many Chern characters v = (r,c 1 ,c 2 ), the corresponding moduli spaces of slope-stable bundles M H (v) when non-empty, are rational, and thereby produce new evidences for a conjecture of Costa and Miró-Roig. As a consequence, we show that the iterated syzygy bundles of Ulrich bundles are dense in these moduli spaces.Objective. The main aim for this article is to study the syzygy bundles of Ulrich bundles. We recall that for any globally generated vector bundle E on a smooth projective variety X , the syzygy bundle (also known in literature as the kernel bundle, the dual span bundle, or the Lazarsfeld-Mukai bundle) M E is defined as the kernel of the evaluation map H 0 (E ) ⊗ O X → E , i.e., it fits into the following exact sequence (1.

In this article we study the deformations of the canonical morphism ϕ : X → P N of irregular surfaces X of general type with at worst canonical singularities, when ϕ is a finite Galois morphism of degree 4 onto a smooth variety of minimal degree Y inside P N . These surfaces satisfy K 2 X = 4p g (X ) − 8, with p g an even integer, p g ≥ 4. They are classified in [GP08] into four distinct families (three, if p g = 4). We show that, when X is general in its family, any deformation of ϕ has degree greater than or equal to 2 onto its image. More interestingly, we prove that, with two exceptions, a general deformation of ϕ is two-to-one onto its image, which is a surface whose normalization is a ruled surface of appropriate genus. We also show that with the exception of one family, the deformations of a general surface X are unobstructed, and consequently, X belongs to a unique irreducible component of the Gieseker moduli space, which we prove is uniruled (the fourth one being product of curves is well-studied). As a consequence we show the existence of infinitely many moduli spaces with uniruled components corresponding to each even p g ≥ 4. Among other things, our results are relevant because they exhibit moduli components such that the degree of the canonical morphism jumps up at proper locally closed subloci. This contrasts with the moduli of surfaces with K 2 X = 2p g −4 (which are double covers of surfaces of minimal degree), studied by Horikawa but is similar to the moduli of curves of genus g ≥ 3.

In this article, we study the moduli of irregular surfaces of general type with at worst canonical singularities satisfying $$K^2 = 4p_g-8$$ K 2 = 4 p g - 8 , for any even integer $$p_g\ge 4$$ p g ≥ 4 . These surfaces also have unbounded irregularity q. We carry out our study by investigating the deformations of the canonical morphism $$\varphi :X\rightarrow {\mathbb {P}}^N$$ φ : X → P N , where $$\varphi $$ φ is a quadruple Galois cover of a smooth surface of minimal degree. These canonical covers are classified in Gallego and Purnaprajna (Trans Am Math Soc 360(10):5489-5507, 2008) into four distinct families, one of which is the easy case of a product of curves. The main objective of this article is to study the deformations of the other three, non trivial, unbounded families. We show that any deformation of $$\varphi $$ φ factors through a double cover of a ruled surface and, hence, is never birational. More interestingly, we prove that, with two exceptions, a general deformation of $$\varphi $$ φ is two-to-one onto its image, whose normalization is a ruled surface of appropriate irregularity. We also show that, with the exception of one family, the deformations of X are unobstructed even though $$H^2(T_X)$$ H 2 ( T X ) does not vanish. Consequently, X belongs to a unique irreducible component of the Gieseker moduli space. These irreducible components are uniruled. As a result of all this, we show the existence of infinitely many moduli spaces, satisfying the strict Beauville inequality $$p_g > 2q-4$$ p g > 2 q - 4 , with an irreducible component that has a proper "quadruple" sublocus where the degree of the canonical morphism jumps up. These components are above the Castelnuovo line, but nonetheless parametrize surfaces with non birational canonical morphisms. The existence of jumping subloci is a contrast with the moduli of surfaces with $$K^2 = 2p_g- 4$$ K 2 = 2 p g - 4 , studied by Horikawa. Irreducible moduli components with a jumping sublocus also present a similarity and a difference to the moduli of curves of genus $$g\ge 3$$ g ≥ 3 , for, like in the case of curves, the degree of the canonical morphism goes down outside a closed sublocus but, unlike in the case of curves, it is never birational. Finally, our study shows that there are infinitely many moduli spaces with an irreducible component whose general elements have non birational canonical morphism and another irreducible component whose general elements have birational canonical map.