Surfaces Of General Type Research Articles

A homology plane of general type can have at most a cyclic quotient singularity

We show that a homology plane of general type has at worst a single cyclic quotient singular point. An example of such a surface with a singular point does exist. We also show that the automorphism group of a smooth contractible surface of general type is cyclic.

A new family of surfaces of general type with $$K^2=7$$ K 2 = 7 and $$p_g=0$$ p g = 0

We construct a new family of smooth minimal surfaces of general type with \(K^2=7\) and \(p_g=0\). We show that a surface in this family has ample canonical divisor and birational bicanonical morphism. We also prove that these surfaces satisfy Bloch’s conjecture.

Mixed quasi-étale surfaces, new surfaces of general type with $$p_g=0$$ and their fundamental group

We call a projective surface $$X$$ mixed quasi-etale quotient if there exists a curve $$C$$ of genus $$g(C)\ge 2$$ and a finite group $$G$$ that acts on $$C\times C$$ exchanging the factors such that $$X=(C\times C)/G$$ and the map $$C\times C \rightarrow X$$ has finite branch locus. The minimal resolution of its singularities is called mixed quasi-etale surface. We study the mixed quasi-etale surfaces under the assumption that $$(C\times C)/G^0$$ has only nodes as singularities, where $$G^0\triangleleft G$$ is the index two subgroup of the elements that do not exchange the factors. We classify the minimal regular surfaces with $$p_g=0$$ whose canonical model is a mixed quasi-etale quotient as above. All these surfaces are of general type and as an important byproduct, we provide an example of a numerical Campedelli surface with topological fundamental group $$\mathbb{Z }_4$$ , and we realize 2 new topological types of surfaces of general type. Three of the families we construct are $$\mathbb{Q }$$ -homology projective planes.

SURFACES OF GENERAL TYPE WITH pg= 1 AND q = 0

We construct a new family of simply connected minimal complex surfaces of general type with <TEX>$p_g$</TEX> = 1, <TEX>$q$</TEX> = 0, and <TEX>$K^2$</TEX> = 3, 4, 5, 6, 8 using a <TEX>$\mathbb{Q}$</TEX>-Gorenstein smoothing theory.

Derived categories of Burniat surfaces and exceptional collections

We construct an exceptional collection $\Upsilon$ of maximal possible length 6 on any of the Burniat surfaces with $K_X^2=6$, a 4-dimensional family of surfaces of general type with $p_g=q=0$. We also calculate the DG algebra of endomorphisms of this collection and show that the subcategory generated by this collection is the same for all Burniat surfaces. The semiorthogonal complement $\mathcal A$ of $\Upsilon$ is an "almost phantom" category: it has trivial Hochschild homology, and $K_0(\mathcal A)=\bZ_2^6$.

Canonical surfaces with big cotangent bundle

Surfaces of general type with positive second Segre number are known to have big cotangent bundle. We give a new criterion ensuring that a surface of general type with canonical singularities has a minimal resolution with big cotangent bundle. This provides many examples of surfaces with negative second Segre number and big cotangent bundle.

ADJOINT LINEAR SYSTEMS ON ALGEBRAIC SURFACES

In this paper, we study the adjoint linear system |KS + L| on a complex surface S of general type. For a nef divisor L on S, we give a numerical criterion for ϕKS+L to be generically finite. A question of Chen and Viehweg [Bicanonical and adjoint linear system on surface of general type, Pacific J. Math.219 (2005) 83–95] is studied and we give some partial evidence for this question to be true.

Burniat-type surfaces and a new family of surfaces with $$p_g = 0, K^2=3$$

The paper is one of a series devoted to the classification, the moduli spaces and the classification of surfaces of general type with \(p_g=0\). Here we generalize a classical construction due to P. Burniat (revised by M. Inoue). Among other results we construct a family of surfaces of general type with \(K_S^2 = 3\), \(p_g(S) = 0\) realizing a new fundamental group of order 16.

Surfaces with pg = q = 2, K2 = 6, and Albanese Map of Degree 2

AbstractWe classify minimal surfaces of general type with pg = q = 2 and K2 = 6 whose Albanese map is a generically finite double cover. We show that the corresponding moduli space is the disjoint union of three generically smooth irreducible components MIa, MIb, MII of dimension 4, 4, 3, respectively.

A complex surface of general type with 𝑝_{𝑔}=0, 𝐾²=2 and 𝐻₁=ℤ/4ℤ

We construct a new minimal complex surface of general type with p g = 0 p_g=0 , K 2 = 2 K^2=2 and H 1 = Z / 4 Z H_1=\mathbb {Z}/4\mathbb {Z} (in fact, π 1 alg = Z / 4 Z \pi _1^{\text {alg}}=\mathbb {Z}/4\mathbb {Z} ), which settles the existence question for numerical Campedelli surfaces with all possible algebraic fundamental groups. The main techniques involved in the construction are a rational blow-down surgery and a Q \mathbb {Q} -Gorenstein smoothing theory.

A simply connected numerical Campedelli surface with an involution

We construct a simply connected minimal complex surface of general type with \(p_g=0\) and \(K^2=2\) which has an involution such that the minimal resolution of the quotient by the involution is a simply connected minimal complex surface of general type with \(p_g=0\) and \(K^2=1\). In order to construct the example, we combine a double covering and \(\mathbb Q \)-Gorenstein deformation. Especially, we develop a method for proving unobstructedness for deformations of a singular surface by generalizing a result of Burns and Wahl which characterizes the space of first order deformations of a singular surface with only rational double points. We describe the stable model in the sense of Kollár and Shepherd-Barron of the singular surfaces used for constructing the example. We count the dimension of the invariant part of the deformation space of the example under the induced \(\mathbb Z /{2}\mathbb Z \)-action.

Remarks on surfaces with $c_1^2 =2\chi -1$ having non-trivial 2-torsion

We shall show that any complex minimal surface of general type with c_1^2 = 2\chi -1 having non-trivial 2-torsion divisors, where c_1^2 and \chi are the first Chern number of a surface and the Euler characteristic of the structure sheaf respectively, has the Euler characteristic \chi not exceeding 4. Moreover, we shall give a complete description for the surfaces of the case \chi =4, and prove that the coarse moduli space for surfaces of this case is a unirational variety of dimension 29. Using the description, we shall also prove that our surfaces of the case \chi = 4 have non-birational bicanonical maps and no pencil of curves of genus 2, hence being of so called non-standard case for the non-birationality of the bicanonical maps.

Burniat surfaces. III: Deformations of automorphisms and extended Burniat surfaces

We continue our investigation of the connected components of the moduli space of surfaces of general type containing the Burniat surfaces, correcting a mistake in part II. We define the family of extended Burniat surfaces with K_S^2 = 4 , resp. 3, and prove that they are a deformation of the family of nodal Burniat surfaces with K_S^2 = 4 , resp. 3. We show that the extended Burniat surfaces together with the nodal Burniat surfaces with K_S^2=4 form a connected component of the moduli space. We prove that the extended Burniat surfaces together with the nodal Burniat surfaces with K_S^2=3 form an irreducible open set in the moduli space. Finally we point out an interesting pathology of the moduli space of surfaces of general type given together with a group of automorphisms G . In fact, we show that for the minimal model S of a nodal Burniat surface (G = (\mathbb Z/2 \mathbb Z)^2) we have \operatorname{Def}(S,G) \neq \operatorname{Def}(S) , whereas for the canonical model X it holds \operatorname{Def}(X,G) =\operatorname{Def}(X) . All deformations of S have a G -action, but there are different deformation types for the pairs (S,G) of the minimal models S together with the G -action, while the pairs (X,G) have a unique deformation type.

On surfaces of general type with q = 5

We prove that a complex surface S with irregularity q(S) = 5 that has no irrational pencil of genus > 1 has geometric genus pg(S) � 8. As a consequence, one is able to classify minimal surfaces S of general type with q(S) = 5 and pg(S) 1 and with the lowest possible geometric genus pg = 2q 3. This gives some evidence for the conjecture that the only irregular surface with no irrational pencil of genus > 1 and pg = 2q 3 is the symmetric product of a genus three curve.

Syzygies of surfaces of general type

We prove new results on projective normality, normal presentation and higher syzygies for a surface of general type $X$ embedded by adjoint line bundles $L_r = K + rB$, where $B$ is a base point free, ample line bundle. Our main results determine the $r$ for which $L_r$ has $N_p$ property. In corollaries, we will relate the bounds on $r$ to the regularity of $B$. Examples in the last section show that several results are optimal.

On the hyperbolicity of surfaces of general type with small c 1 2

Surfaces of general type with positive second Segre number s2: = c12 − c2 > 0 are known by the results of Bogomolov to be algebraically quasi-hyperbolic, that is, with finitely many rational and elliptic curves. These results were extended by McQuillan in his proof of the Green–Griffiths conjecture for entire curves on such surfaces. In this work, we study hyperbolic properties of minimal surfaces of general type with minimal c12 known as Horikawa surfaces. In principle, these surfaces should be the most difficult case for the above conjecture as illustrated by the quintic surfaces in ℙ3. Using orbifold techniques, we exhibit infinitely many irreducible components of the moduli of Horikawa surfaces whose very generic member has no rational curves or is algebraically hyperbolic. Moreover, we construct explicit examples of algebraically hyperbolic and quasi-hyperbolic orbifold Horikawa surfaces.

Canonical double covers of minimal rational surfaces and the non-existence of carpets

This article delves into the relation between the deformation theory of finite morphisms to projective space and the existence of ropes, embedded in projective space, with certain invariants. We focus on the case of canonical double covers X of a minimal rational surface Y, embedded in PN by a complete linear series, and carpets on Y, canonically embedded in PN. We prove that these canonical double covers always deform to double covers and that canonically embedded carpets on Y do not exist. This fact parallels the results known for hyperelliptic canonical morphisms of curves and canonical ribbons, and the results for K3 double covers of surfaces of minimal degree and Enriques surfaces and K3 carpets. That canonical double covers of minimal rational surfaces should deform to double covers is not a priori obvious, for the invariants of most of these surfaces lie on or above the Castelnuovo line; thus, in principle, deformations of such covers could have birational canonical maps. In fact, many canonical double covers of non-minimal rational surfaces do deform to birational canonical morphisms.We also map the region of the geography of surfaces of general type corresponding to the surfaces X and we compute the dimension of the irreducible moduli component containing [X]. In certain cases we exhibit some interesting moduli components parameterizing surfaces S with the same invariants as X but with birational canonical map, unlike X.

Quotients of Fano surfaces

Fano surfaces parametrize the lines of smooth cubic threefolds. In this paper, we study their quotients by some of their automorphism sub-groups. We obtain in that way some interesting surfaces of general type.

Simply connected surfaces of general type in positive characteristic via deformation theory

Algebraically simply connected surfaces of general type with p_g=q=0 and 1\le K^2\le 4 in positive characteristic (with one exception in K^2=4) are presented by using a Q-Gorenstein smoothing of two-dimensional toric singularities, a generalization of Lee-Park's construction in the field of complex numbers to the positive characteristic case, and Grothendieck's specialization theorem for the fundamental group.

Quotients of products of curves, new surfaces with p g = 0 and their fundamental groups.

We construct many new surfaces of general type with $q=p_g = 0$ whose canonical model is the quotient of the product of two curves by the action of a finite group $G$, constructing in this way many new interesting fundamental groups which distinguish connected components of the moduli space of surfaces of general type. We indeed classify all such surfaces whose canonical model is singular (the smooth case was classified in an earlier work). As an important tool we prove a structure theorem giving a precise description of the fundamental group of quotients of products of curves by the action of a finite group $G$.