EXTENDING HYPERELLIPTIC K3 SURFACES, AND GODEAUX SURFACES WITH π<sub>1</sub>= ℤ/2

Coughlan, Stephen

doi:10.4134/jkms.j150307

Cited by 7 publications

(16 citation statements)

References 8 publications

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“…We note that these particular surfaces are relevant for the description of the moduli space of Z/2-Godeaux surfaces, which was recently proved [DR20] to be a rational irreducible variety of dimension 8, finishing with the previous attempts e.g. [CD80,CCO94,Co16]. For K 2 = 3 much less is known, the torsion of the Picard group has order at most 4, and the ones with torsion Z/3 are described in [M03].…”

Section: Introductionsupporting

confidence: 67%

Rational configurations in K3 surfaces and simply-connected $p_g=1$ surfaces for $K^2=1,2,3,4,5,6,7,8,9$

Reyes¹,

Urzúa²

2021

Preprint

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We prove the existence of (20 − 2K 2 )-dimensional families of simply-connected surfaces with ample canonical class, p g = 1, and 1 ≤ K 2 ≤ 9, and we study the relation with configurations of rational curves in K3 surfaces via Q-Gorenstein smoothings. Our surfaces with K 2 = 7 and K 2 = 9 are the first surfaces known in the literature, together with the existence of a 4dimensional family for K 2 = 8. Contents 1. Introduction 1 2. Wahl degenerations of p g = 1 surfaces 5 3. Rational configurations in K3 surfaces 13 4. Simply-connected p g = 1 surfaces for 2 ≤ K 2 ≤ 9 17 4.1. K 2 = 2 19 4.2. K 2 = 3 20 4.3. K 2 = 4 21 4.4. K 2 = 5 22 4.5. K 2 = 6 23 4.6. K 2 = 7 25 4.7. K 2 = 8 26 4.8. K 2 = 9 28 4.9. More surfaces for each 2 ≤ K 2 ≤ 9 29 References 35

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Section: Introductionsupporting

confidence: 67%

Rational configurations in K3 surfaces and simply-connected $p_g=1$ surfaces for $K^2=1,2,3,4,5,6,7,8,9$

Reyes¹,

Urzúa²

2021

Preprint

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“…Coughlan [Cou16] has obtained a family of Z/2-Godeaux surfaces (i.e. with torsion group Z/2) depending on 8 parameters.…”

Section: Introductionmentioning

confidence: 99%

$\mathbb Z/2$-Godeaux surfaces

Dias¹,

Rito²

2020

Preprint

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We compute explicit equations for all (universal coverings of) Godeaux surfaces with torsion group Z/2. We show that their moduli space is irreducible and rational of dimension 8. 2020 MSC: 14J29[SS20] claiming the construction of an 8-dimensional family of simply connected Godeaux surfaces, but without obtaining a full classification).Catanese and Debarre [CD89] showed that the étale double covers of Z/2-Godeaux surfaces have hyperelliptic canonical curve and birational bicanonical map onto an octic in P 3 , and they did a general study of its canonical ring. That octic is given by the determinant of a certain matrix α.In this paper we continue their work. Using an idea from Miles Reid [Rei90], we get more precise information about α by looking first to its restriction to the case of the canonical curve, then extending to the surface. Then we give an algorithm for the computation of all such matrices, from which we obtain equations for the étale double covers of all Z/2-Godeaux surfaces. We show that their moduli space is irreducible of dimension 8, which implies that the topological fundamental group of Z/2-Godeaux surfaces is also Z/2.We note that our method is not brute force computation: for the main algorithm, the calculations used only 32 MB of RAM memory, and took 85 seconds on a low-end computer.Recently two special Z/2-Godeaux surfaces have appeared in the literature: a (Z/3) 2 -quotient of a fake projective plane, constructed by Borisov and Fatighenti [BF20], which has 4 cusp singularities; a degree 6 quotient of the so-called Cartwright-Steger surface, given by Borisov-Yeung [BY20], which has 3 cusp singularities and a certain configuration of rational curves. As an exercise, we give the coordinates of these surfaces in our family.All computations are implemented with Magma [BCP97], and can be found in some arXiv ancillary files. In particular, using the files 5_Verifications_alpha_i_c_j.txt one can choose any surface in the family and compute its invariants and singular set. Results from Catanese-DebarreWe collect here some results from the paper [CD89] that will be used throughout the text.Let S be the étale double cover of a numerical Godeaux surface with torsion group Z/2, and denote the corresponding involution by σ. The invariants of S are K 2 = 2, p g = 1, q = 0. Define the canonical ring of S as R = ∞ n=0 H 0 (S, nK S ), and let A = C[x, y 1 , y 2 , y 3 ] be the C-graded algebra with deg(x) = 1, deg(y i ) = 2.

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“…Although the above ordering is a local one, one can check that those polynomials are still in the ideal I Γ . We note that these 6 polynomials were also determined in [C16,Cor. 4.3].…”

Section: Type IV Unprojectionmentioning

confidence: 96%

“…Proposition 3.1. [C16,Prop. 3.3] The projection from the node P ∈ T ⊂ P(2 4 , 3 4 , 4) gives a complete intersection…”

Section: Extending Hyperelliptic K3 Surfacesmentioning

confidence: 99%

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On degenerations of $\mathbb Z/2$-Godeaux surfaces

Dias¹,

Rito²,

Urzúa³

2020

Preprint

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We compute equations for the Coughlan's family in [C16] of Godeaux surfaces with torsion Z/2, which we call Z/2-Godeaux surfaces, and we show that it is (at most) 7 dimensional. We classify non-rational KSBA degenerations W of Z/2-Godeaux surfaces with one Wahl singularity, showing that W is birational to particular either Enriques surfaces, or D2,n elliptic surfaces, with n = 3, 4 or 6. We present examples for all possibilities in the first case, and for n = 3, 4 in the second.

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EXTENDING HYPERELLIPTIC K3 SURFACES, AND GODEAUX SURFACES WITH π₁= ℤ/2

Cited by 7 publications

References 8 publications

Rational configurations in K3 surfaces and simply-connected $p_g=1$ surfaces for $K^2=1,2,3,4,5,6,7,8,9$

Rational configurations in K3 surfaces and simply-connected $p_g=1$ surfaces for $K^2=1,2,3,4,5,6,7,8,9$

$\mathbb Z/2$-Godeaux surfaces

On degenerations of $\mathbb Z/2$-Godeaux surfaces

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EXTENDING HYPERELLIPTIC K3 SURFACES, AND GODEAUX SURFACES WITH π1= ℤ/2

Cited by 7 publications

References 8 publications

Rational configurations in K3 surfaces and simply-connected $p_g=1$ surfaces for $K^2=1,2,3,4,5,6,7,8,9$

Rational configurations in K3 surfaces and simply-connected $p_g=1$ surfaces for $K^2=1,2,3,4,5,6,7,8,9$

$\mathbb Z/2$-Godeaux surfaces

On degenerations of $\mathbb Z/2$-Godeaux surfaces

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EXTENDING HYPERELLIPTIC K3 SURFACES, AND GODEAUX SURFACES WITH π₁= ℤ/2