Algebraic Surfaces Of General Type Research Articles

Surfaces of General Type with Maximal Picard Number Near the Noether Line

Abstract The first published non-trivial examples of algebraic surfaces of general type with maximal Picard number are due to Persson, who constructed surfaces with maximal Picard number on the Noether line $K^{2}=2\chi -6$ for every admissible pair $(K^{2},\chi )$ such that $\chi \not \equiv 0 \ \text {mod}\ 6$. In this note, given a non-negative integer $k$, algebraic surfaces of general type with maximal Picard number lying on the line $K^{2}=2\chi -6+k$ are constructed for every admissible pair $(K^{2},\chi )$ such that $\chi \geq 2k+10$. These constructions, obtained as bidouble covers of rational surfaces, not only allow to fill in Persson’s gap on the Noether line, but also provide infinitely many new examples of algebraic surfaces of general type with maximal Picard number above the Noether line.

Some surfaces with canonical maps of degrees 10, 11, and 14

AbstractIn this note, we present examples of complex algebraic surfaces of general type with canonical maps of degrees 10, 11, and 14. They are constructed as quotients of a product of two Fermat septics using certain free actions of the group .

A new infinite family of irregular algebraic surfaces with canonical map of degree 8

In this note, we construct an unlimited family of irregular algebraic surfaces of general type with canonical map of degree 8, irregularity 1, and arbitrarily large geometric genus such that the image of the canonical map is not a surface of minimal degree.

Algebraic surfaces of general type with

In this paper, we study minimal algebraic surfaces of general type with Since the case has been studied by Horikawa, Zucconi, and Bauer, we always assume here. We prove that for such a surface, the canonical map is generically finite of degree 1 or 2. If the degree of the canonical map is 2, then the canonical image is a ruled surface, moreover, for any fixed there exists such a regular surface. If the degree of the canonical map is 1, then the surface is regular (i.e. q = 0) and its canonical linear system is base point free or has one simple base point. In the case where the canonical map is birational and the canonical linear system is base point free, we study the property of its canonical image and prove that for any fixed there exists such a surface.

ℤ3-actions on Horikawa surfaces

Minimal algebraic surfaces of general type [Formula: see text] such that [Formula: see text] are called Horikawa surfaces. In this note, [Formula: see text]-actions on Horikawa surfaces are studied. The main result states that given an admissible pair [Formula: see text] such that [Formula: see text], all the connected components of Gieseker’s moduli space [Formula: see text] contain surfaces admitting a [Formula: see text]-action. On the other hand, the examples considered allow us to produce normal stable surfaces that do not admit a [Formula: see text]-Gorenstein smoothing. This is illustrated by constructing non-smoothable normal surfaces in the KSBA-compactification [Formula: see text] of Gieseker’s moduli space [Formula: see text] for every admissible pair [Formula: see text] such that [Formula: see text]. Furthermore, the surfaces constructed belong to connected components of [Formula: see text] without canonical models.

$${\mathbb {Z}}_2^2$$-actions on Horikawa surfaces

Minimal algebraic surfaces of general type X such that \(K^2_X=2\chi ({\mathcal {O}}_X)-6\) or \(K^2_X=2\chi ({\mathcal {O}}_X)-5\) are called Horikawa surfaces. In this note we study \({\mathbb {Z}}^2_2\)-actions on Horikawa surfaces. The main result is that all the connected components of Gieseker’s moduli space of canonical models of surfaces of general type with invariants satisfying these relations contain surfaces with \({\mathbb {Z}}^2_2\)-actions.

Algebraic surfaces of general type with $$p_g=q=1$$ p g = q = 1 and genus 2 Albanese fibrations

In this paper, we study algebraic surfaces of general type with $p_g=q=1$ and genus 2 Albanese fibrations. We first study the examples of surfaces with $p_g=q=1, K^2=5$ and genus 2 Albanese fibrations constructed by Catanese using singular bidouble covers of $\mathbb{P}^2$. We prove that these surfaces give an irreducible and connected component of $\mathcal{M}_{1,1}^{5,2}$, the Gieseker moduli space of surfaces of general type with $p_g=q=1, K^2=5$ and genus 2 Albanese fibrations. Then by constructing surfaces with $p_g=q=1,K^2=3$ and a genus 2 Albanese fibration such that the number of the summands of the direct image of the bicanonical sheaf (under the Albanese map) is 2, we give a negative answer to a question of Pignatelli.

A surface of maximal canonical degree

It is known since the 1970s from a paper of Beauville that the degree of the rational canonical map of a smooth projective algebraic surface of general type is at most 36. Though it has been conjectured that a surface with optimal canonical degree 36 exists, the highest canonical degree known earlier for a minimal surface of general type was 16 by Persson. The purpose of this paper is to give an affirmative answer to the conjecture by providing an explicit surface.

On algebraic surfaces of general type with negative

We prove that for any prime number $p\geqslant 3$, there exists a positive number $\unicode[STIX]{x1D705}_{p}$ such that $\unicode[STIX]{x1D712}({\mathcal{O}}_{X})\geqslant \unicode[STIX]{x1D705}_{p}c_{1}^{2}$ holds true for all algebraic surfaces $X$ of general type in characteristic $p$. In particular, $\unicode[STIX]{x1D712}({\mathcal{O}}_{X})>0$. This answers a question of Shepherd-Barron when $p\geqslant 3$.

Topology and dynamics of laminations in surfaces of general type

We study Riemann surface laminations in complex algebraic surfaces of general type. We focus on the topology and dynamics of minimal sets of holomorphic foliations and on Levi-flat hypersurfaces. We begin by providing various examples. Then our first result shows that Anosov Levi-flat hypersurfaces do not embed in surfaces of general type. This allows one to classify the possible Thurston’s geometries carried by Levi-flat hypersurfaces in surfaces of general type. Our second result establishes that minimal sets in surfaces of general type have a large Hausdorff dimension as soon as there exists a simply connected leaf. For both results, our methods rely on ergodic theory: we use harmonic measures and Lyapunov exponents.

Surfaces of general type with K2=2χ−1

We classify minimal algebraic surfaces of general type having K2=2χ−1 and χ≥7. Such surfaces are regular with canonical map of degree one or two. If pg≥13, then the surface is a genus-two fibration; otherwise we use the canonical map to describe these surfaces as birational either to the canonical image or to a double cover of a rational surface.

The Tate Conjecture for a family of surfaces of general type with p g = q = 1 and K 2 = 3

We prove a big monodromy result for a smooth family of complex algebraic surfaces of general type, with invariants $p_g=q=1$ and $K^2=3$, that has been introduced by Catanese and Ciliberto. This is accomplished via a careful study of degenerations. As corollaries, when a surface in this family is defined over a finitely generated extension of $\Bbb{Q}$, we verify the semisimplicity and Tate conjectures for the Galois representation on the middle $\ell$-adic cohomology of the surface.

On the canonical ring of curves and surfaces

Let C be a curve (possibly non reduced or reducible) lying on a smooth algebraic surface. We show that the canonical ring $${ R(C, \omega_C)=\bigoplus_{k\geq 0} H^0(C, {\omega_C}^{\otimes k})}$$ is generated in degree 1 if C is numerically four-connected, not hyperelliptic and even (i.e. with ω C of even degree on every component). As a corollary we show that on a smooth algebraic surface of general type with p g (S) ≥ 1 and q(S) = 0 the canonical ring R(S, K S ) is generated in degree ≤ 3 if there exists a curve $${C \in |K_S|}$$ numerically three-connected and not hyperelliptic.

Constructions of Kähler–Einstein metrics with negative scalar curvature

We show that on Kähler manifolds with negative first Chern class, the sequence of algebraic metrics introduced by Tsuji converges uniformly to the Kähler–Einstein metric. For algebraic surfaces of general type and orbifolds with isolated singularities, we prove a convergence result for a modified version of Tsuji’s iterative construction.

Algebraic Surfaces of General Type with Small c21 in Positive Characteristic

AbstractWe establish Noether’s inequality for surfaces of general type in positive characteristic. Then we extend Enriques’ and Horikawa’s classification of surfaces on the Noether line, the so-called Horikawa surfaces. We construct examples for all possible numerical invariants and in arbitrary characteristic, where we need foliations and deformation techniques to handle characteristic 2. Finally, we show that Horikawa surfaces lift to characteristic zero.

Diffeomorphism of simply connected algebraic surfaces

In this paper we show that even in the case of simply connected minimal algebraic surfaces of general type, deformation and differentiable equivalence do not coincide. Exhibiting several simple families of surfaces which are not deformation equivalent, and proving their diffeomorphism, we give a counterexample to a weaker form of the speculation DEF = DIFF of R. Friedman and J. Morgan, i.e., in the case where (by M. Freedman’s theorem) the topological type is completely determined by the numerical invariants of the surface. We hope that the methods of proof may turn out to be quite useful to show diffeomorphism and indeed symplectic equivalence for many important classes of algebraic surfaces and symplectic 4-manifolds.

Numerical Godeaux surfaces with an involution

Minimal algebraic surfaces of general type with the smallest possible invariants have geometric genus zero andK2=1K^2=1and are usually callednumerical Godeaux surfaces. Although they have been studied by several authors, their complete classification is not known. In this paper we classify numerical Godeaux surfaces with an involution, i.e. an automorphism of order 2. We prove that they are birationally equivalent either to double covers of Enriques surfaces or to double planes of two different types: the branch curve either has degree 10 and suitable singularities, originally suggested by Campedelli, or is the union of two lines and a curve of degree 12 with certain singularities. The latter type of double planes are degenerations of examples described by Du Val, and their existence was previously unknown; we show some examples of this new type, also computing their torsion group.

Signature of relations in mapping class groups and non-holomorphic Lefschetz fibrations

We introduce the notion of signature for relations in mapping class groups and show that the signature of a Lefschetz fibration over the 2-sphere is the sum of the signatures for basic relations contained in its monodromy. Combining explicit calculations of the signature cocycle with a technique of substituting positive relations, we give some new examples of non-holomorphic Lefschetz fibrations of genus 3 , 4 3, 4 and 5 5 which violate slope bounds for non-hyperelliptic fibrations on algebraic surfaces of general type.

Minimal algebraic surfaces of general type with $c^2_1 = 3, p_g = 1 \text{ and } q = 0$, which have non-trivial 3-torsion divisors

We shall give a concrete description of minimal algebraic surfaces $X$’s defined over $\mathbb{C}$ of general type with the first chern number 3, the geometric genus 1 and the irregularity 0, which have non-trivial 3-torsion divisors. Namely, we shall show that the fundamental group is isomorphic to $\mathbb{Z}/3$, and that the canonical model of the universal cover is a complete intersection in $\mathbb{P}^{4}$ of type (3, 3).

The classification of double planes of general type with K2=8 and pg=0

We study minimal double planes of general type with K 2 =8 and p g =0, namely pairs ( S , σ ), where S is a minimal complex algebraic surface of general type with K 2 =8 and p g =0, and σ is an automorphism of S of order 2 such that the quotient S / σ is a rational surface. We prove that S is a free quotient ( F × C )/ G , where C is a curve, F is an hyperelliptic curve, G is a finite group that acts faithfully on F and C , and σ is induced by the automorphism τ ×Id of F × C , τ being the hyperelliptic involution of F . We describe all the F , C , and G that occur: in this way we obtain 5 families of surfaces with p g =0 and K 2 =8, of which we believe only one was previously known. Using our classification we are able to give an alternative description of these surfaces as double covers of the plane, thus recovering a construction proposed by Du Val. In addition, we study the geometry of the subset of the moduli space of surfaces of general type with p g =0 and K 2 =8 that admit a double plane structure.