Deformations of hyperelliptic and generalized hyperelliptic polarized varieties

Abstract

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.

K3 carpets on minimal rational surfaces and their smoothings

“…Calabi Yau varieties are higher dimensional analogues of K3 surfaces. In [BGG20], they show that there are no Calabi Yau double structures on projective bundles and P N , in sharp contrast with the results on K3 surfaces in this article. This together with results on deformations in [BGG20] show that deformations of generalized hyperelliptic Calabi-YauY n-fold is again a generalized hyperelliptic n-fold.…”

“…The results in [BGG20] show that there are no higher dimensional analogues of the above results. They introduce the notion of generalized hyperelliptic varieties that unifies various results on deformations of double covers.…”

“…In a recent article [BGG20], it is shown that there are no higher dimensional analogues of these results. In [BGG20] authors introduce the notion of generalized hyperelliptic varieties.…”

“…In a recent article [BGG20], it is shown that there are no higher dimensional analogues of these results. In [BGG20] authors introduce the notion of generalized hyperelliptic varieties. It has been shown that the deformations of generalized hyperelliptic polarized Calabi-Yau varieties of dimension ≥ 3 are again generalized hyperelliptic varieties ([BGG20], Theorem 4.5).…”

“…Multiple structures arise in a variety of contexts in algebraic geometry. For example, they arise in the study of vector bundles and linear series (see [BM85], [HV85], [M81]), as well as in deformation and moduli problems (see [BGG20], [GGP10], [GGP08a]). One of the interesting features of K3 carpets is that the hyperplane section of K3 carpets supported on scrolls are canonical ribbons.…”