The Classification of Surfaces with pg = q = 0 Isogenous to a Product of Curves (original) (raw)
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.
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
On the canonical map of some surfaces isogenous to a product
Local and Global Methods in Algebraic Geometry, 2018
We construct several families (indeed, connected components of the moduli space) of surfaces S of general type with p g = 5, 6 whose canonical map has image Σ of very high degree, d = 48 for p g = 5, d = 56 for p g = 6. And a connected component of the moduli space consisting of surfaces S with K 2 S = 40, p g = 4, q = 0 whose canonical map has always degree ≥ 2, and, for the general surface, of degree 2 onto a canonical surface Y with K 2 Y = 12, p g = 4, q = 0. The surfaces we consider are SIP 's, i.e. surfaces S isogenous to a product of curves (C 1 × C 2 )/G; in our examples the group G is elementary abelian, G ∼ = (Z/m) k . We also establish some basic results concerning the canonical maps of any surface isogenous to a product, basing on elementary representation theory. Our methods and results are a first step towards answering the question of existence of SIP 's S with p g = 6, q = 0 whose canonical map embeds S as a surface of degree 56 in P 5 .
D.: Beauville surfaces with abelian Beauville group
2016
A Beauville surface is a rigid surface of general type arising as a quotient of a product of curves C1, C2 of genera g1, g2 ≥ 2 by the free action of a finite group G. In this paper we study those Beauville surfaces for which G is abelian (so that G ∼ = Z 2 n with gcd(n, 6) = 1 by a result of Catanese). For each such n we are able to describe all such surfaces, give a formula for the number of their isomorphism classes and identify their possible automorphism groups. This explicit description also allows us to observe that such surfaces are all defined over Q.
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.
Product-Quotient Surfaces: new invariants and algorithms
arXiv (Cornell University), 2013
In this article we suggest a new approach to the systematic, computeraided construction and to the classification of product-quotient surfaces, introducing a new invariant, the integer γ, which depends only on the singularities of the quotient model X = (C 1 × C 2)/G. It turns out that γ is related to the codimension of the subspace of H 1,1 generated by algebraic curves coming from the construction (i.e., the classes of the two fibers and the Hirzebruch-Jung strings arising from the minimal resolution of singularities of X). Profiting from this new insight we developped and implemented an algorithm which constructs all regular product-quotient surfaces with given values of γ and geometric genus in the computer algebra program MAGMA. Being far better than the previous algorithms, we are able to construct a substantial number of new regular product-quotient surfaces of geometric genus zero. We prove that only two of these are of general type, raising the number of known families of product-quotient surfaces of general type with genus zero to 75. This gives evidence to the conjecture that there is an effective bound Γ(p g , q) ≥ γ (cf. Conjecture 4.4).