A family of surfaces with pg=q=2,K2=7 and Albanese map of degree 3 (original) (raw)

A note on a family of surfaces with p_g=q=2andandandK^2=7$$

Bollettino dell'Unione Matematica Italiana, 2021

We study a family of surfaces of general type with p g = q = 2 and K 2 = 7, originally constructed by C. Rito in [Rit18]. We provide an alternative construction of these surfaces, that allows us to describe their Albanese map and the corresponding locus M in the moduli space of the surfaces of general type. In particular we prove that M is an open subset, and it has three connected components, two dimensional, irreducible and generically smooth.

Note on a family of surfaces with pg=q=2p_g=q=2pg=q=2 and K2=7K^2=7K2=7

arXiv (Cornell University), 2020

Surfaces with p g = q =3

manuscripta mathematica, 2002

In this paper it is proved that a complete algebraic surface of general type with p g = q = 3, without irrational pencil of genus bigger than one is birationally equivalent to the two-symmetric product of a curve of genus 3. This result completes the classification of the surfaces with p g = q = 3. The main tools are the Lefschetz theorem and the use of the paracanonical system on the surface.

A new family of surfaces with pg=0p_g=0pg=0 and K2=3K^2=3K2=3

arXiv (Cornell University), 2003

Let S be a minimal complex surface of general type with p g = 0 such that the bicanonical map ϕ of S is not birational and let Z be the bicanonical image. In [M. Mendes Lopes, R. Pardini, Enriques surfaces with eight nodes, Math. Zeit. 241 4 (2002), 673-683] it is shown that either: i) Z is a rational surface, or ii) K 2 S = 3, ϕ is a degree two morphism and Z is birational to an Enriques surface. Up to now no example of case ii) was known. Here an explicit construction of all such surfaces is given. Furthermore it is shown that the corresponding subset of the moduli space of surfaces of general type is irreducible and uniruled of dimension 6.

Some new surfaces with pg=q=0p_g = q = 0pg=q=0

2003

Motivated by a question by D. Mumford : can a computer classify all surfaces with pg=0p_g = 0pg=0 ? we try to show the complexity of the problem. We restrict it to the classification of the minimal surfaces of general type with pg=0,K2=8p_g = 0, K^2 = 8pg=0,K2=8 which are constructed by the Beauville construction, namely, which are quotients of

A Class of Rational Surfaces in P

2003

In this paper, we obtain a complete classification of all rational surfaces embedded in P4 so that all their exceptional curves are lines. These surfaces are exactely the rational surfaces shown by I.Bauer to project isomorphicaly from P5 from one of their points, although no a priori reason is known why such a surface should be projectable in this way.

The Moduli Space of Even Surfaces of General

2016

Even surfaces of general type with K 2 = 8, pg = 4 and q = 0 were found by Oliverio [Ol05] as complete intersections of bidegree (6, 6) in a weighted projective space P(1, 1, 2, 3, 3). In this article we prove that the moduli space of even surfaces of general type with K 2 = 8, pg = 4 and q = 0 consists of two 35-dimensional irreducible components intersecting in a codimension one subset (the first of these components is the closure of the open set considered by Oliverio). All the surfaces in the second component have a singular canonical model, hence we get a new example of a generically nonreduced moduli space. Our result gives a posteriori a complete description of the half-canonical rings of the above even surfaces. The method of proof is, we believe, the most interesting part of the paper. After describing the graded ring of a cone we are able, combining the explicit description of some subsets of the moduli space, some deformation theoretic arguments, and finally some local algebra arguments, to describe the whole moduli space. This is the first time that the classification of a class of surfaces can only be done using moduli theory: up to now first the surfaces were classified, on the basis of some numerical inequalities, or other arguments, and later on the moduli spaces were investigated. 2 FABRIZIO CATANESE, WENFEI LIU, AND ROBERTO PIGNATELLI surfaces which we mentioned above are surfaces with 4 ≤ K 2 ≤ 7; by the work of Ciliberto and Catanese, [C81] and [Cat99], existence is known for each 4 ≤ K 2 ≤ 28. Irregular surfaces with p g = 4 were later investigated in [CS02]: in this case K 2 ≥ 8 since, by [De82], one has K 2 ≥ 2p g for irregular surfaces 1 ; while K 2 ≥ 12 if the canonical map has degree 1. Surfaces with p g = 4 and K 2 = 4 were classified by Noether and Enriques, but it took the work of Horikawa and Bauer ([Ho76a, Ho76b, Ho78, B01]) to finish the classification of the surfaces with p g = 4 and 4 ≤ K 2 ≤ 7 (necessarily regular). These are 'essentially' classified, in the sense that the moduli space is shown to be a union of certain (explicitly described) locally closed subsets: but there is missing complete knowledge of the incidence structure of these subsets of the moduli space. We refer to the survey [BCP06b] for a good account of the range 4 ≤ K 2 ≤ 7, and to [Cat97] for a previous more general survey (containing the construction of several new examples). Minimal surfaces with K 2 = 8, p g = 4, q = 0 have been the object of further work by several authors [C81, CFM97, Ol05]. The surfaces constructed by Ciliberto have a birational canonical map, are not even, and have a trivial torsion group H 1 (S, Z) (unlike the ones considered in [CFM97]); the ones constructed by Oliverio are simply connected (see [D82]), and they are even (meaning that the canonical divisor is divisible by two: i.e., the second Stiefel Whitney class w 2 (S) = 0, equivalently, the intersection form is even). Therefore, for K 2 = 8, p g = 4, q = 0 there are at least three different topological types [Ol05, Remark 5.4], contrasting the situation for (minimal) surfaces with p g = 4, K 2 ≤ 7 which, when they have the same K 2 , are homeomorphic to each other. Recently Bauer and the third author [BP09] classified minimal surfaces with K 2 = 8, p g = 4, q = 0 whose canonical map is composed with an involution (while examples with canonical map of degree three are given in [MP00]). Their work shows that the moduli space of minimal surfaces with K 2 = 8, p g = 4, q = 0 has at least four irreducible components: and a new fifth one is described in the present paper. Therefore the classification of minimal surfaces with K 2 = 8, p g = 4, q = 0 seems a very challenging problem, yet not completely out of reach. The present article provides a first step in this direction, classifying all the even surfaces and completely describing the corresponding subset M ev 8,4,0 of the moduli space. This is our main result: denote by M ev 8,4,0 the moduli space of even surfaces of general type with K 2 = 8, p g = 4 and q = 0. We show that M ev 8,4,0 , which a priori consists of several connected components of the whole moduli space M 8,4,0 , is indeed a single connected component of the moduli space M 8,4,0. Oliverio [Ol05] found out that, if |K S | is base point free (this condition determines an open set of the moduli space) the half-canonical ring R(S, L) realizes the canonical model X of the surface X as a (6, 6) complete intersection in P(1 2 , 2, 3 2). Since conversely such complete intersections having at worst Du Val singularities (i.e., rational double points) yield such canonical models, one gets as a result that this open set is an irreducible unirational open set of dimension 35 in the moduli space M ev 8,4,0 , hence 1 That the case K 2 = 8, pg = 4 and q = 1 actually occurs is shown by the family of double covers of the product E × P 1 , where E is an elliptic curve and the branch divisor has numerical type (4, 6).

A family of surfaces with pg=q=2,K2=7 and Albanese map of degree 3 (original) (raw)

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