On the Extendability of Projective Surfaces and a Genus Bound for Enriques-Fano Threefolds

Abstract

We introduce a new technique, based on Gaussian maps, to study the possibility, for a given surface, to lie on a threefold as a very ample divisor with given normal bundle. We give several applications, among which one to surfaces of general type and another one to Enriques surfaces. For the latter we prove that any threefold (with no assumption on its singularities) having as hyperplane section a smooth Enriques surface (by definition an Enriques-Fano threefold) has genus g ≤ 17 (where g is the genus of its smooth curve sections). Moreover we find a new Enriques-Fano threefold of genus 9 whose normalization has canonical but not terminal singularities and does not admit Q-smoothings.Besides the mentioned applications, in the present article we will concentrate most of our attention on the case of Enriques-Fano threefolds: in analogy with Fano threefolds where an anticanonical divisor is a K3 surface, we define Definition 1.3. An Enriques-Fano threefold is an irreducible three-dimensional variety X ⊂ P N having a hyperplane section S that is a smooth Enriques surface, and such that X is not a cone over S. We will say that X has genus g if g is the genus of its general curve section.Fano himself, in a 1938 article [Fa], claimed a classification of such threefolds, but his proof contains several gaps. Conte and Murre [CM] were the first to remark that an Enriques-Fano threefold must have some isolated singularities, typical examples of which are quadruple points with tangent cone the cone over the Veronese surface. Filling out some of the gaps in [Fa] and making some special assumptions on the singularities, Conte and Murre recovered some of the results of Fano, but not enough to give a classification, nor to bound the numerical invariants. On the opposite extreme, with the strong assumption that the Enriques-Fano threefold is a quotient of a smooth Fano threefold by an involution (this corresponds to having only cyclic quotient terminal singularities), a list was given by Bayle [Ba, Thm.A] and Sano [Sa, Thm.1.1], by using the classification of smooth Fano threefolds and studying which of them have such involutions.Moreover, by the results of Minagawa [Mi, MainThm2], any Enriques-Fano threefold with at most terminal singularities admits a Q-smoothing, that is [Mi, R1], it appears as central fiber of a small deformation over the 1-parameter unit disk, such that a general fiber has only cyclic quotient terminal singularities. This, together with the results of Bayle and Sano, gives then the bound g ≤ 13 for Enriques-Fano threefolds with at most terminal singularities.Bayle and Sano recovered all of the examples of Enriques-Fano threefolds given by Fano and Conte-Murre. As these were the only known examples, it has been conjectured for some time now that this list is complete or, at least, that the genus is bounded, in analogy with the celebrated genus bound for smooth Fano threefolds [I1, I2, Sh2].In Section 16, we will show that the list of known Enriques-Fano threefolds of Fano, Conte-Murre, Bayle and Sano is not...

Moduli of curves on Enriques surfaces

“…(Indeed, when k = 2, since (S, H) and (S, H + K S ) are both general in E • 6,2 we have that h 1 (T S (−H)) and h 1 (T S (−(H + K S )) are equal, and (27) implies they are both equal to 2. )…”

“…We conclude the section by explaining how to use our results (without using [27,38]) to bound the families of Enriques-Fano threefolds having the property that their Enriques sections are general in moduli, meaning that the family of polarized Enriques sections obtained from the family dominates the moduli space E of Enriques surfaces. Corollary 4.10.…”

“…We recall that a projective variety V ⊂ P N is said to be k-extendable if there exists a projective variety W ⊂ P N +k , different from a cone, such that V = W ∩ P N (transversely). The question of k-extendability of Enriques surfaces is still open, although it is proved in [38,27] that N 17 is a necessary condition for 1-extendability, and terminal threefolds having Enriques surfaces as hyperplane sections have been classified in [4,41,30]. This result is sharp in the case of 1-extendability: The general members of the moduli spaces of Corollary 1.2 indeed occur as hyperplane sections of threefolds different from cones, cf.…”

“…Corollary 6.5. We remark that the proof of Corollary 1.2 is independent from, and much simpler than, the results in [38,27], but it needs the technical assumption about rational curves, which can probably be avoided, at the expense of adding more cases, cf. Remark 6.1.…”

О Трехмерных Многообразиях Фано С Каноническими Горенштейновыми Особенностями

“…Пусть U -многообразие Фано-Энриквеса такое, что −K U ≡ H для некоторого дивизора Картье H. Согласно [13; лемма 2.2] g := 1 2 H 3 + 1 -целое число; этород U . Проблема получения точной оценки для рода многообразий Фано-Энриквеса изучалась в [17] и [18]. В [17] была установлена точная оценка g 17 в предположении, что особенности U изолированные.…”