# A simply connected numerical Godeaux surface with ample canonical class

@article{Dolgachev1997ASC, title={A simply connected numerical Godeaux surface with ample canonical class}, author={Igor Dolgachev and Caryn Werner}, journal={arXiv: Algebraic Geometry}, year={1997} }

We prove that a recent construction of a numerical Godeaux surface due to P. Craighero and R. Gattazzo is simply connected, and show how to realize their construction as a double plane. By proving that the surface contains no (-2)-curves, we obtain that this is the first example of a simply connected surface with vanishing geometric genus and ample canonical class.

## 36 Citations

### The Craighero–Gattazzo surface is simply connected

- MathematicsCompositio Mathematica
- 2017

We show that the Craighero–Gattazzo surface, the minimal resolution of an explicit complex quintic surface with four elliptic singularities, is simply connected. This was conjectured by Dolgachev and…

### Canonical rings of Gorenstein stable Godeaux surfaces

- Mathematics
- 2016

Extending the description of canonical rings from Reid (J Fac Sci Univ Tokyo Sect IA Math 25(1):75–92, 1978) we show that every Gorenstein stable Godeaux surface with torsion of order at least 3 is…

### Bloch's conjecture on surfaces of general type with an involution

- Mathematics
- 2017

In this short note we prove that the Bloch's conjecture holds for a surface of general type of numerical Godeaux type or some class of numerical Campedelli type, with geometric genus zero equipped…

### Degenerations Of Godeaux Surfaces And Exceptional Vector Bundles

- Mathematics
- 2013

A recent construction of Hacking relates the classification of stable vector bundles on a surface of general type with $p_g = 0$ and the boundary of the moduli space of deformations of the surface.…

### Numerical Godeaux surfaces with an involution

- Mathematics
- 2005

Minimal algebraic surfaces of general type with the smallest possible invariants have geometric genus zero and K 2 = 1 and are usually called numerical Godeauz surfaces. Although they have been…

### Gorenstein stable Godeaux surfaces

- Mathematics
- 2016

We classify Gorenstein stable numerical Godeaux surfaces with worse than canonical singularities and compute their fundamental groups.

### Gorenstein stable Godeaux surfaces

- MathematicsSelecta Mathematica
- 2017

We classify Gorenstein stable numerical Godeaux surfaces with worse than canonical singularities and compute their fundamental groups.

### Chow groups of conic bundles in $\mathbb P^5$ and the Generalised Bloch's conjecture

- Mathematics
- 2017

Consider the Fano surface of a conic bundle embedded in $\mathbb P^5$. Let $i$ denote the natural involution acting on this surface. In this note we provide an obstruction to the identity action of…

### On the class of projective surfaces of general type

- Mathematics
- 2017

Let S be a smooth complex projective surface of general type, H be a very ample divisor on S and m(S, H) be the class of (S, H). In this paper, we study a lower bound for m(S, H)− 3H2 and we improve…

### Generalised Bloch's conjecture on surfaces of general type with involution

- Mathematics
- 2017

In this short note we prove that an involution on certain examples of surfaces of general type with $p_g=0$, acts as identity on the Chow group of zero cycles of the relevant surface.

## References

SHOWING 1-10 OF 11 REFERENCES

### On the scalar curvature of Einstein manifolds

- Mathematics
- 1997

We show that there are high-dimensional smooth compact manifolds which admit pairs of Einstein metrics for which the scalar curvatures have oppo- site signs. These are counter-examples to a…

### A surface of general type with pg=q=0,K2=1

- Mathematics
- 1994

We construct a surface of general type with invariants \( \chi = K^2 = 1 \) and torsion group \( \Bbb{Z}/{2} \). We use a double plane construction by finding a plane curve with certain…

### On Campedelli branch loci

- MathematicsANNALI DELL UNIVERSITA DI FERRARA
- 1997

SuntoSi costruiscono nuove curve piane di ordine dieci dotate di sei punti [3, 3] che non giacciono su una conica e curve di ordine dieci dotate di cinque punti [3, 3] e di un punto quadruplo ancora…

### Campedelli versus Godeaux, in “Problems in the theory of surfaces and their classification

- (Cortona,
- 1988

### Campedelli versus Godeaux, in " Problems in the theory of surfaces and their classification (Cortona, 1988)

- Campedelli versus Godeaux, in " Problems in the theory of surfaces and their classification (Cortona, 1988)
- 1991