Feb 19, 2014 - Page 1 .... Figure 1. The geography of minimal and Gorenstein stable surfaces. Our results ... The first author was supported by the Bi...

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arXiv:1307.1999v2 [math.AG] 19 Feb 2014

Abstract. We study the geography of Gorenstein stable log surfaces and prove two inequalities for their invariants: the stable Noether inequality and the P2 inequality. By constructing examples we show that all invariants are realised except possibly some cases where the inequalities become equalities.

Contents 1. Introduction 2. Preliminaries 3. Riemann–Roch and the P2 -inequality 4. Noether inequality 5. Examples References

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1. Introduction The classification of algebraic surfaces has been a subject of interest in algebraic geometry ever since the foundational work of the Italian school at the beginning of last century and the complexity of this endeavour led Castelnuovo and Enriques to their saying: “If curves have been made by God, then surfaces are the devil’s mischief.” For surfaces of general type, one aspect is their geography, that is, the question about general restrictions on their invariants and the construction of surfaces realising all possible invariants. The compactification Ma,b of Gieseker’s moduli space of canonical models of sur2 and b = χ(O ) parametrises stable surfaces (see faces of general type with a = KX X Definition 2.1). That such surfaces should be the correct higher-dimensional analogue of stable curves was first suggested by Kollár and Shepherd-Barron [KSB88]; formidable technical obstacle delayed the actual construction of the moduli space for several decades [Kol14]. While it is still true that the Gieseker moduli space Ma,b is an open subset of Ma,b , the complement is no longer a divisor as in the moduli space of stable curves: there can be additional irreducible components and for some invariants Ma,b might be empty while Ma,b is not. This simply means that the invariants of some stable surfaces cannot be realised by surfaces of general type. It is actually quite natural to consider also stable log surfaces (Definition 2.1), where we allow a reduced boundary. As a first step beyond the classical case we prove two fundamental inequalities for Gorenstein stable log surfaces. 2010 Mathematics Subject Classification. 14J10, 14J29. Key words and phrases. stable surface, stable log surface, geography of surfaces. 1

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P2 -inequality (Theorem 3.3) — Let (X, ∆) be a connected Gorenstein stable log surface. Then χ(X, ωX (∆)) = χ(OX (−∆)) ≥ −(KX + ∆)2 , ⊗2 and equality holds if and only if ∆ = 0 and P2 (X) = h0 (X, ωX ) = 0.

Stable log Noether inequality (Theorem 4.1, Corollary 4.3) — Let (X, ∆) be a connected Gorenstein stable log surface. Then pg (X, ∆) = h0 (X, ωX (∆)) ≤ (KX + ∆)2 + 2, χ(X, ωX (∆)) ≤ (KX + ∆)2 + 2, and the first inequality is strict if ∆ = 0. In both cases, the strategy is to use well known results in the normal respectively smooth case. For the P2 -inequality this is relatively straightforward (apart from a small issue with adjunction) while for the Noether inequality one needs to control the combinatorics of the glueing process carefully. In contrast to the case of minimal surfaces of general type, most of the possible invariants are realised by a simple combinatorial construction explained in Section 5.1. All these examples are locally smoothable but global smoothability may or may not occur (see Section 5.1.7). To put these results in context, let us discuss the known restrictions on invariants for some classes of surfaces with empty boundary. In the following, X will always denote a surface of the specified type. In all cases KX is an ample Q-Cartier divisor 2 > 0, which may however be a rational number if X is not Gorenstein. so we have KX Minimal surfaces of general type: The following well-known inequalities are satisfied: Euler characteristic: χ(OX ) > 0. 2 + 2 (or K 2 ≥ 2χ(O ) − 6). Noether inequality: pg (X) ≤ 21 KX X X 2 ≤ 9χ(O ). Bogomolov–Miyaoka–Yau inequality: KX X A proof of these inequalities and references showing that almost all possible invariants are known to be realised can be found in [BHPV04, Ch. VII]. Normal stable surfaces: It has been proved by Blache that also in this case χ(OX ) > 0 [Bla94, Thm. 2]. There is an analogue of the Bogomolov-MiyaokaYau inequality (see [Lan03] and references therein) that can be stated in terms of the orbifold Euler-characteristic 2 KX ≤ 3eorb (X).

However, the orbifold Euler-characteristic is not invariant under deformation so it is less suited to the moduli point of view. We show that both the classical Noether-inequality and the classical Bogomolov–Miyaoka–Yau inequality fail for normal Gorenstein stable surfaces in Section 5.2. General case: It is known that 2 KX | X stable surface is a DCC set, bounded below by 1/1726 [AM04, Kol94] but our understanding is far from complete. For example, it is very difficult to bound the index for surfaces with fixed invariants. We expect that a kind of Noether-inequality holds also in this case, see Remark 4.2.

GEOGRAPHY OF GORENSTEIN STABLE LOG SURFACES 2 KX

P2 = 0

BMY

Noether

3

stable Noether

∅ χ(OX ) Gorenstein stable surfaces

minimal surfaces

Constructed in Sect. 5.1

Figure 1. The geography of minimal and Gorenstein stable surfaces Our results for Gorenstein stable surfaces without boundary are illustrated in Figure 1, where we also mark the points where an explicit example has been constructed. The stable log Noether inequality is sharp and we give a partial characterisation of surfaces on the stable log Noether line in Corollary 4.10. On the other hand, we have some evidence to believe that there are no Gorenstein stable surfaces with 2 ≥ 3. χ(OX ) − 2 = KX Surfaces with negative χ(OX ) are a bit more mysterious. It can be shown that 2 = 1 and P (X) = 0 [FPR13] so one might wonder if there is no surface with KX 2 always P2 (X) > 0. If both the above speculations on the sharpness of the inequalities turn out to be true then every possible invariant is realised by the examples in Section 5.1. In the last section we give some further examples that illustrate some of the obstacles in working with stable surfaces. For example, the classical approach to prove Noether’s inequality would be to look at the image of the canonical map. We show that for a stable surface this image may be disconnected or not equidimensional which made a different approach necessary. We also include an example of a 1-dimensional 2 = 9, fake fake projective planes, family of stable surfaces with χ(OX ) = 1 and KX which confirms that there is in general no direct relation between stable surfaces and minimal surfaces with the same invariants. Acknowledgements: We are grateful to Fabrizio Catanese, Marco Franciosi, Christian Liedtke, Michael Lönne and Rita Pardini for interesting discussions about this project. János Kollár sent us a preliminary version of [Kol13]. Matthias Schütt suggested to construct a family of fake fake projective planes. Both authors were supported by DFG via the second author’s Emmy-Noether project and partially via SFB 701. The first author was supported by the Bielefelder Nachwuchsfonds. 1.1. Notations and conventions. We work exclusively with schemes of finite type over the complex numbers. • The singular locus of a scheme X will be denoted by Xsing .

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• A surface is a reduced, projective scheme of pure dimension two but not necessarily irreducible or connected. • A curve is a purely 1-dimensional scheme that is Cohen–Macaulay. A curve is not assumed to be reduced, irreducible or connected; its arithmetic genus is pa (C) = 1 − χ(OC ). • By abuse of notation we sometimes do not distinguish a divisor D and the associated divisorial sheaf OX (D); this is especially harmless for Cartier divisors. 2. Preliminaries In this section we recall some necessary notions as well as constructions that we need throughout the text. Most of these are available in all dimensions, but for our purpose it suffices to focus on the case of surfaces. Our main reference is [Kol13, Sect. 5.1–5.3]. 2.1. Stable log surfaces. Let X be a demi-normal surface, that is, X satisfies S2 and at each point of codimension one X is either regular or has an ordinary ¯ → X the normalisation of X. The conductor double point. We denote by π : X ideal H omOX (π∗ OX¯ , OX ) is an ideal sheaf in both OX and OX¯ and as such defines ¯ ⊂ X, ¯ both reduced and of pure codimension 1; we often subschemes D ⊂ X and D refer to D as the non-normal locus of X. Let ∆ be a reduced curve on X whose support does not contain any irreducible ¯ in the normalisation is well defined. component of D. Then the strict transform ∆ Definition 2.1 — We call a pair (X, ∆) as above a log surface; ∆ is called the (reduced) boundary.1 A log surface (X, ∆) is said to have semi-log-canonical (slc) singularities if it satisfies the following conditions: (i) KX + ∆ is Q-Cartier, that is, m(KX + ∆) is Cartier for some m ∈ Z>0 ; the minimal such m is called the (global) index of (X, ∆). ¯ D ¯ + ∆) ¯ has log-canonical singularities. (ii) The pair (X, The pair (X, ∆) is called stable log surface if in addition KX + ∆ is ample. A stable surface is a stable log surface with empty boundary. By abuse of notation we say (X, ∆) is a Gorenstein stable log surface if the index is equal to one, i.e., KX + ∆ is an ample Cartier divisor. ¯ → D on the Since X has at most double points in codimension one the map π : D conductor divisors is generically a double cover and thus induce a rational involution ¯ Normalising the conductor loci we get an honest involution τ : D ¯ν → D ¯ ν such on D. ν ν ¯ /τ . that D = D To state the next result we need the notion of different, which is the correction term in the adjunction formula on a log surface. Definition 2.2 ([Kol13, Definition 4.2]) — Let (X, ∆) be a log surface and B a reduced curve on X that does not contain any irreducible component of the nonnormal locus D. Suppose ωX (∆ + B)[m] is a line bundle for some positive integer m. Then, denoting by B ν the normalisation of B, the different Diff B ν (∆) is the uniquely determined Q-divisor on B ν such that mDiff B ν (∆) is integral and the residue map 1In general one can allow rational coefficients in ∆, but we will not use this here.

GEOGRAPHY OF GORENSTEIN STABLE LOG SURFACES

5

induces an isomorphism [m] ωX (∆ + B)[m] |B ν ∼ = ωB ν (mDiff B ν (∆)).

Theorem 2.3 ([Kol13, Thm. 5.13]) — Associating to a log-surface (X, ∆) the triple ¯ D ¯ + ∆, ¯ τ: D ¯ν → D ¯ ν ) induces a one-to-one correspondence (X, ¯ ¯ ¯ log-canonical pair (X, D + ∆) ( stable log ) ¯ +∆ ¯ ample, with KX¯ + D ¯ ¯ surfaces ↔ (X, D, τ ) ¯ν → D ¯ ν an involution . τ : D (X, ∆) s.th. Diff D¯ ν (∆) is τ -invariant. An important consequence, which allows to understand the geometry of stable log surfaces from the normalisation, is that ¯ X (1)

¯ D

π

π

X

D

ν

¯ν D /τ

Dν

is a pushout diagram. Definition 2.4 — Let (X, ∆) be a stable log surface. We call pg (X, ∆) = h0 (X, ωX (∆)) = h2 (X, OX (−∆)) the geometric genus of (X, ∆) and q(X, ∆) = h1 (X, ωX (∆)) = h1 (X, OX (−∆)) the irregularity of (X, ∆). If ∆ is empty we omit it from the notation. Note that in both cases for the second equality we have used [LR12, Lem. 3.3] and duality. We will want to relate the invariants of a stable log surface with the invariants of the normalisation. ¯ ∆). ¯ Proposition 2.5 — Let (X, ∆) be a stable log surface with normalisation (X, 2 2 ¯ + ∆) ¯ and χ(OX ) = χ(O ¯ ) + χ(OD ) − χ(O ¯ ). Then (KX + ∆) = (KX¯ + D X D Proof. The first part is clear. For the second note that the conductor ideal defines ¯ on X ¯ and the non-normal locus D on X. In particular, π∗ O ¯ (−D) ¯ = ID and D X additivity of the Euler characteristic for the two sequences ¯ → O ¯ → O ¯ → 0, 0 → OX¯ (−D) X D ¯ → OX → OD → 0 0 → π∗ O ¯ (−D) X

gives the claimed result.

To compare the irregularity and the geometric genus is more subtle. We state the following general result in the Gorenstein case for simplicity. Proposition 2.6 ([Kol13, Prop. 5.8]) — If (X, ∆) is a Gorenstein log surface then ¯ ω ¯ (D ¯ + ∆)) ¯ is the subspace of those sections s such that π ∗ H 0 (X, ωX (∆)) ⊂ H 0 (X, X ¯ ν , ω ¯ ν (Diff ¯ ν (∆))) is τ -anti-invariant. the residue of s in H 0 (D D

D

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Remark 2.7 — We need the following consequence of the above proposition. Assume that X = X1 ∪ X2 is the union of two (not necessarily irreducible) surfaces such that the conductor is a smooth curve and let C = X1 ∩ X2 . With ∆i = ∆|X the residue i maps define homomorphisms rXi : H 0 (Xi , KXi + ∆i + C) → H 0 (C, (KXi + ∆i + C)|C ) ∼ = H 0 (C, KC + Diff C (∆)). Then there is a fibre product diagram of vector spaces H 0 (X, KX + ∆)

H 0 (X1 , KX1 + ∆1 + C) rX1

H 0 (X2 , KX2 + ∆2 + C)

−rX2

.

H 0 (C, KC + Diff C (∆))

By abuse of notation we also write H 0 (X, KX + ∆) = H 0 (X1 , KX1 + ∆1 + C) ×C H 0 (X2 , KX2 + ∆2 + C). 2.2. Gorenstein slc singularities and semi-resolutions. Normalising a deminormal surface looses all information on the glueing in codimension one. Often it is better to work on a simpler but still non-normal surface. Definition 2.8 — A surface X is called semi-smooth if every point of X is either smooth or double normal crossing or a pinch point2. A morphism of demi-normal surfaces f : Y → X is called a semi-resolution if the following conditions are satisfied: (i) Y is semi-smooth; (ii) f is an isomorphism over the semi-smooth open subscheme of X; (iii) f maps the singular locus of Y birationally onto the non-normal of X. A semi-resolution f : Y → X is called minimal if no (−1)-curve is contracted by f , that is, there is no exceptional curve E such that E 2 = KY · E = −1. Semi-resolutions always exist and one can also incorporate a boundary [Kol13, Sect. 10.5]. Remark 2.9 (Classification of Gorenstein slc singularities) — Semi-log-canonical surface singularities have been classified in terms of their resolution graphs, at least for reduced boundary [KSB88]. Let x ∈ (X, ∆) be a Gorenstein slc singularity with minimal log semi-resolution f : Y → X. Then it is one of the following (see [KM98, Ch. 4], [Kol13, Sect. 3.3], and [Kol12, 17]): Gorenstein lc singularities, ∆ = 0: In this case x ∈ X is smooth, a canonical singularity, or a simple elliptic respectively cusp singularitiy. For the latter the resolution graph is a smooth elliptic curve, a nodal rational curve, or a cycle of smooth rational curves (see also [Lau77] and [Rei97, Ch. 4]). Gorenstein lc singularities, ∆ 6= 0: Since the boundary is reduced, ∆ has at most nodes. If ∆ is smooth so is X because of the Gorenstein assumption. If ∆ has a node at x then x is a smooth point of X or (X, ∆) is a general hyperplane section of a finite quotient singularity. In the minimal log resolution the dual graph of the exceptional curves is • − c1 − · · · − cn − •

(ci ≥ 1)

2A local model for the pinch point in A3 is given by the equation x2 + yz 2 = 0.

GEOGRAPHY OF GORENSTEIN STABLE LOG SURFACES

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Figure 2. Semi-resolution and normalisation of a degenerate cusp glued from two planes and two A1 -singularities. ¯ X

Resolution graph D1

D2

2 1

1 2

D4

D3

π X

D1

D2

Y f

−2 −1

−1 −2

D3

D4

where ci represents a smooth rational curve of self-intersection −ci and each • represents a (local) component of the strict transform of ∆. If ci = 1 for some i then n = 1 and ∆ is a normal crossing divisor in a smooth surface. non-normal Gorenstein slc singularities, ∆ = 0: We describe the dual graph of the f -exceptional divisors over x: analytically locally X consists of k irreducible components, on each component we have a resolution graph as in the previous item, and these are glued together where the components intersect. In total we have a cycle of smooth rational curve. For example, the normalisation of the hypersurface singularity Tp,∞,∞ = {xyz + xp = 0} (p ≥ 3) consists of a plane and an Ap−2 singularity. The resolution graph of the semi-resolution is obtained by attaching • − 2 − · · · − 2 − • and • − 1 − • to a circle. A more graphical example is given in Figure 2. non-normal Gorenstein slc singularities, ∆ 6= 0: The difference to the previous case is that the local components are now glued in a chain and the ends of the chain intersect the strict transform of the boundary. In this case X itself might not even be Q-Gorenstein. 2.3. Intersection product and the hat transform. Mumford’s Q-valued intersection product can be extended to demi-normal surfaces as long as the curves do not have common components with the conductor, that is, are almost Cartier divisors in the sense of [Har94]. Definition 2.10 — Let X be a demi-normal surface and let A, B be Q-Weil-divisors on X whose support does not contain any irreducible component of the conductor ¯ be the strict transforms of A and B on the normalisation X. ¯ Then D. Let A¯ and B we define an intersection pairing ¯ AB := A¯B

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WENFEI LIU AND SÖNKE ROLLENSKE

where the right hand side is Mumford’s intersection pairing. One should consider the resulting intersection numbers with care if both A and B are not Cartier. For example, the intersection number of curves on different irreducible components of X is always zero in this definition even if the geometric intersection is non-empty. We also need to recall the hat-transform of a curve in an slc surface constructed in [LR12, Appendix A]. Proposition 2.11 — Let X be a demi-normal surface and B ⊂ X be a curve which does not contain any irreducible component of the conductor D. Let f : Y → X be the minimal semi-resolution. ˆY = B, for all ˆY on Y such that f∗ B (i) There exists a unique Weil divisor B ˆY E ≤ 0 and B ˆY is minimal with this exceptional divisors E of f we have B property. (ii) If X has slc singularities then ˆ¯) − 2 + B ˆ¯ D¯ 2pa (B) − 2 ≤ 2pa (B Y

Y

Y

ˆ ¯ is the strict transform of B ˆY in the normalisation. where B Y 3. Riemann–Roch and the P2 -inequality Theorem 3.1 (Riemann–Roch for Cartier divisors) — Let X be a demi-normal surface and L be a Cartier divisor on X. Then 1 χ(OX (L)) = χ(OX ) + L(L − KX ). 2 ¯ ¯ ⊂X ¯ the conductor. We tensor Proof. Let π : X → X be the normalisation and D the structure sequence of the double locus, 0 → ID → OX → OD → 0,

(2) with OX (L) and get (3)

χ(OX (L)) = χ(ID ⊗ OX (L))) + χ(OX (L)|D ).

¯ and using projection formula π∗ π ∗ OX (L)(−D) ¯ = ID ⊗ OX (L) Pulling (2) back to X we have χ(ID ⊗ OX (L)) = χ(π ∗ OX (L)) − χ(π ∗ OX (L)|D¯ ) Adding this to (3) and applying the Riemann–Roch formula for Cartier divisors on ¯ and Proposition 2.5 we get normal surfaces ([Bla95]), Riemann–Roch on D and D, χ(OX (L)) = χ(π ∗ OX (L)) − χ(π ∗ OX (L)|D¯ ) + χ(OX (L)|D ) 1 = χ(OX¯ ) + π ∗ L(π ∗ L − KX¯ ) − χ(π ∗ OX (L)|D¯ ) + χ(OX (L)|D ) 2 1 ¯ + D) ¯ = χ(OX¯ ) + π ∗ L(π ∗ L − KX¯ − D 2 ¯ + (χ(OD ) + deg L| ) − (χ(OD¯ ) + π ∗ LD) D 1 = χ(OX¯ ) + χ(OD ) − χ(OD¯ ) + L(L − KX ) 2 1 = χ(OX ) + L(L − KX ). (by Proposition 2.5) 2 This is the claimed formula.

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Corollary 3.2 — Let (X, ∆) be a Gorenstein stable log surface. Then for m ≥ 2 Pm (X, ∆) = h0 (ωX (∆)⊗ m ) = χ(OX ) +

m m(m − 1) (KX + ∆)2 + (KX + ∆)∆ 2 2

Proof. Apply Theorem 3.1 to L = m(KX + ∆) and use that higher cohomology vanishes by [LR12, Corollary 3.4]. Theorem 3.3 (P2 -inequality) — Let (X, ∆) be a Gorenstein stable log surface. Then χ(ωX (∆)) = χ(OX (−∆)) ≥ −(KX + ∆)2 , ⊗2 and equality holds if and only if ∆ = 0 and P2 (X) = h0 (X, ωX ) = 0.

Proof. If ∆ = 0 then by Corollary 3.2 we have 2 0 ≤ P2 (X) = χ(OX ) + KX ,

which gives the claimed formula. Now suppose ∆ 6= 0. We must prove χ(ωX (∆)) + (KX + ∆)2 > 0. We begin by applying Theorem 3.1 to ωX (∆) and ωX (∆)⊗ 2 and taking the difference of the resulting formulas, which gives (4)

1 χ(ωX (∆)) + (KX + ∆)2 = χ(ωX (∆)⊗ 2 ) − (KX + ∆)∆. 2

To calculate the right hand side we use the exact sequence 0 → OX (2KX + ∆) → OX (2KX + 2∆) → O∆ (2KX + 2∆) → 0. Together with Riemann–Roch on ∆ we obtain χ(ωX (∆)⊗ 2 ) = χ(OX (2KX + ∆)) + χ(O∆ ) + 2(KX + ∆)∆, where we used that for a Cartier divisor the degree on a curve coincides with the intersection product. Thus (4) becomes 1 χ(ωX (∆))+(KX +∆)2 = (KX +∆)∆+χ(OX (2KX +∆))+ χ(O∆ ) + (KX + ∆)∆ . 2 By [LR12, Corollary 3.4] we have H i (X, 2KX + ∆) = 0 for i > 0 and hence χ(X, 2KX + ∆) = h0 (X, 2KX + ∆) ≥ 0. Since KX + ∆ is ample and we assumed ∆ 6= 0 the first summand is strictly positive. It remains to control the last summand. This is accomplished by the following estimate, which seems rather trivial, recall however that we are working on a non-normal surface. Claim — (KX + ∆)∆ + 2χ(O∆ ) ≥ 0. Proof of the Claim. Let f : Y → X be the minimal semi-resolution and π : Y¯ → Y the normalisation. Let Z be the exceptional divisor of f and Z¯ its strict transform ˆ ¯ ⊂ Y¯ be the strict transform of the hat transform of ∆ (see in Y¯ . Let further ∆ Y Proposition 2.11) which, by adjunction on Y¯ satisfies (5)

ˆ ¯. ˆ ¯ )∆ − 2χ(O∆ ) ≤ (KY¯ + DY¯ + ∆ Y Y

Going through the cases in the classification of Gorenstein slc singularities with ˆ ¯ explicitly. In fact, ∆ ˆ¯ = non-empty boundary (Remark 2.9) one can compute ∆ Y Y

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WENFEI LIU AND SÖNKE ROLLENSKE

∆Y¯ + Z¯ 0 , where Z¯ 0 consists of the reduced connected components of Z¯ that intersect ∆Y¯ .3 So ˆ ¯ )∆ ˆ ¯ = (K ¯ + D ¯ + ∆ ¯ + Z¯ 0 )∆ ˆ¯ (KY¯ + DY¯ + ∆ Y Y Y Y Y Y ¯ ˆ = (K ¯ + D ¯ + ∆ ¯ + Z)∆ ¯ Y ∗ ∗

Y

Y

Y

ˆ¯ = π f (KX + ∆)∆ Y = (KX + ∆)∆. Combining this with (5) finishes the proof of the claim.

Remark 3.4 — Assume X is a Gorenstein stable surface with P2 (X) = 0. By the vanishing results in [LR12, Sect. 2] we have a surjection H 0 (2KX ) H 0 (2KX |D ), so the latter space has to be zero as well. Thus the degree of 2KX |C has to be small, more precisely, by [LR12, Lem. 4.7, Lem. 4.8] the non-normal locus cannot be nodal and X pa (D) − 1 ≥ deg 2KX |D ≥ 4pa (D) − 4 + 2 (2 − µp (D)), p∈Dsing

where µp (D) is the multiplicity of D at p. This imposes strong restrictions on the geometry of D. In fact, the classification results obtained in [FPR13] show that there is no Goren2 = 1 and P (X) = 0. So we suspect that the P stein stable surface with KX 2 2 inequality might not be sharp. 4. Noether inequality In this section we prove the analogue of Noether’s inequality for Gorenstein stable log surfaces. Theorem 4.1 (Stable log Noether inequality) — Let (X, ∆) be a connected Gorenstein stable log surface (with reduced boundary ∆). Then pg (X, ∆) = h0 (X, ωX (∆)) ≤ (KX + ∆)2 + 2. and the inequality is strict if ∆ = 0. Remark 4.2 — The inequality is sharp for pairs, see for example the list of normal log surfaces in Proposition 4.6. We will give a partial characterisation for log surfaces on the stable log Noether line in Corollary 4.10. For surfaces without boundary the strict inequality is also sharp: there are smooth 2 = 1 and p (X) = 2. However, we believe that there are Horikawa surfaces with KX g 2 ≥ 3 holds. no Gorenstein stable surfaces X such that pg (X) − 1 = KX If we drop the Gorenstein condition there are normal stable surfaces X such that 2 + 2 [TZ92, Example 1.8]. The weaker inequality p (X) ≤ pK 2 q + 2 pg (X) > KX g X might be a working hypothesis for the general case. Corollary 4.3 — Let (X, ∆) be a connected Gorenstein stable log surface. Then χ(X, ωX (∆)) ≤ (KX + ∆)2 + 2. 3To get this simple description it is important to work with the minimal semi-resolution instead of the minimal log semi-resolution. If we blow up a node of ∆ which is a smooth point of X then the resulting (−1)-curve occurs with multiplicity 2 in the hat transform.

GEOGRAPHY OF GORENSTEIN STABLE LOG SURFACES

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Proof. Note that χ(X, ωX (∆)) = h0 (X, ωX (∆))−h1 (X, ωX (∆))+h2 (X, ωX (∆)). We have h2 (X, ωX (∆)) = h0 (X, OX (−∆)) which vanishes if ∆ 6= 0 and is 1-dimensional if ∆ = 0. The corollary follows from this and Theorem 4.1.

4.1. Set-up for the proof. Consider the minimal semi-resolution (Y, ∆Y ) of (X, ∆) and the respective normalisations: Y¯

DY¯ ∪ ∆Y¯

f¯

η

DY ∪ ∆Y

Y

¯ X π

f

X.

where ∆Y (resp. ∆Y¯ ) is the strict transform of ∆X in Y (resp. Y¯ ). Our approach is to compute as much as possible on the disjoint union of smooth surfaces Y¯ which F we decompose into irreducible components as Y¯ = ki=1 Y¯i . Irreducible components ¯ i = f¯(Yi ) of the other spaces involved will be numbered correspondingly, that is, X ¯ ¯ ¯ and for a divisor E the part contained in Yi will be denoted by Ei . The non-normal locus of Y is a smooth curve DY and the double cover ηD : DY¯ → DY corresponds to a line-bundle L on DY such that L⊗ 2 is the line bundle associated to the branch points of ηD . A Cartier divisor Z is defined via ωY (∆Y + Z) = f ∗ ωX (∆); it consists of a curve of arithmetic genus 1 for each simple elliptic singularity, cusp or degenerate cusp of X and a chain of rational curves over each the intersection point of ∆ and D. ¯ With these The strict transform of Z in the normalisation Y¯ will be denoted by Z. notations one can check ¯ η ∗ (KY + ∆Y + Z) = KY¯ + DY¯ + ∆Y¯ + Z, ¯ | , η ∗ (KY + ∆Y + Z)|D ¯ = KDY¯ + (∆Y¯ + Z) D¯ Y

Y

and (KX + ∆)2 = η ∗ f ∗ (KX + ∆)2 (6)

¯ 2 = (KY¯ + DY¯ + ∆Y¯ + Z) X = (KY¯i + DY¯ , i + ∆Y¯ , i + Z¯i )2 . i

To relate h0 (X, ωX (∆)) to the geometry of Y¯ we first note that h0 (X, ωX (∆)) = h0 (Y, f ∗ ωX (∆)) = h0 (Y, ωY (∆Y + Z)) by the projection formula. The exact sequence 0 → OY → η∗ OY¯ → L−1 → 0

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WENFEI LIU AND SÖNKE ROLLENSKE

tensored with ωY (∆Y + Z) gives the vertical exact sequence: 0

H 0 (X, ωX (∆)) (f ◦η)∗ Res

¯ H 0 (Y¯ , ωY¯ (DY¯ + ∆Y¯ + Z))

¯ H 0 (DY¯ , ωDY¯ (∆Y¯ + Z))

ρ

H 0 (DY , L−1 ⊗ ωY (∆Y + Z)|D )

H 0 (DY , L−1 ⊗ ωY (∆Y + Z)|D )

Y

Y

and the map ρ factors through the residue map to the conductor divisor. Thus we have X h0 (Y¯i , ω ¯ (D ¯ + ∆ ¯ + Z¯i )) − dim im ρ. (7) H 0 (X, ωX (∆)) = Yi

Y ,i

Y ,i

i

¯ by the τ -invariant subspace In fact, im ρ is the quotient of H 0 (DY¯ , ωDY¯ (∆Y¯ + Z)) by Proposition 2.6. Looking at the equations (6) and (7) our claim is trivial unless there is a component Yi such that h0 (Y¯i , ωY¯i (DY¯ ,i + ∆Y¯ ,i + Z¯i )) is bigger than (KY¯i + DY¯ ,i + ∆Y¯ ,i + Z¯i )2 . If there are such components we have to make sure that the image of ρ is big enough, that is, enough sections do not descend from the normalisation to Y . This will be done by studying the residue map to single components of DY¯ . Before adressing these questions in the next subsection we state a Lemma that picks out those components of the boundary is DY¯ + ∆Y¯ + Z¯ on the normalisation that (possibly) are contained in the conductor. ¯ Then C¯ is Lemma 4.4 — Let C¯ be an irreducible component of DY¯ + ∆Y¯ + Z. ¯ a component of DY¯ + ∆Y¯ if and only if ωY¯ (DY¯ + Z)|C¯ is ample. The line bundle ¯ restricted on the components of Z¯ is trivial. ωY¯ (DY¯ + ∆Y¯ + Z) 4.2. Normal pairs with pg (W, Λ) > (KW + Λ)2 . In this section we study normal pairs with large pg and also lower bounds for the rank of the restriction map to components of the boundary on which the log-canonical divisor is positive. We first recall the following facts extracted from [Sak80, TZ92]. Proposition 4.5 — Let (W, Λ) be a Gorenstein log canonical pair with reduced boundary such that KW + Λ is big and nef. Then pg (W, Λ) ≤ (KW + Λ)2 + 2. Proposition 4.6 — Let (W, Λ) be a Gorenstein log canonical pair such that with f → W be the minimal reduced boundary such that KW + Λ is ample. Let f : W ∗ 2 e = f Λ. If pg (W, Λ) = (KW + Λ) + 2 then the pair (W f , Λ) e is one resolution and Λ of the following: (i) (P2 , nodal quartic curve), (ii) (P2 , nodal quintic curve), f = Fe , Λ e a nodal curve in |3C0 + (2e + k + 2)F | with k ≥ 1. (iii) W f e (iv ) W = Fe , Λ a nodal curve in |3C0 + (2e + 2)F | with e 6= 0.

GEOGRAPHY OF GORENSTEIN STABLE LOG SURFACES

13

Here Fe = P(OP1 ⊕ OP1 (−e)) → P1 denotes a Hirzebruch-surface with section C0 such that C02 = −e and fibre F . A lc pair (W 0 , Λ0 ) such that there is a birational morphism g : W 0 → W with 0 + Λ0 = g ∗ (K KW W + Λ) big and nef and W as above is called pg -extremal pair. Remark 4.7 — Note that if (W 0 , Λ0 ) is pg -extremal then |KW 0 + Λ0 | defines a morphism whose image is the log-canonical model. In particular, each component of the boundary is either contracted to a point or embedded. Lemma 4.8 — Let (W 0 , Λ0 ) be a pg -extremal pair and let C1 , C2 be components of Λ0 such that (KW 0 + Λ0 )Ci > 0. Then the following holds: (i) r(C1 ) = dim im Res : H 0 (W 0 , KW 0 + Λ0 ) → H 0 (C1 , (KW 0 + Λ0 )|C1 ) ≥ 2. If equality holds then C1 is mapped to a line by the map ψ associated to the linear system |KW 0 + Λ0 |. (ii) If C1 and C2 are two different components then r(C1 + C2 ) = dim im Res : H 0 (W 0 , KW 0 + Λ0 ) → H 0 ((KW 0 + Λ0 )|C1 ∪C2 ) ≥ 3. Proof. We may assume that (W 0 , Λ0 ) is one of the smooth surfaces listed in Proposition 4.6. In the first three cases ψ is an embedding while in the fourth case ψ is the minimal resolution of of the cone over a rational normal curve: it contracts the section C0 (see [Har77, Ch. V] for the case of Fe ). In the latter case, the section satisfies (KW 0 + Λ0 )C0 = 0. Therefore ψ is birational when restricted any of the Ci and C1 and C2 do not have the same image. Looking only at one component C1 gives r(C1 ) ≥ 2 with equality if and only if C1 is mapped to a line, thus (i). 0 0 Two (different) curves in Ppg (W ,Λ )−1 span a projective space of dimension at least two, which gives at least three sections in the restriction, i.e., r(C1 + C2 ) ≥ 3. Lemma 4.9 — Let (W, Λ) be a log canonical pair with W smooth and KW + Λ big and nef. If pg (W, Λ) = (KW + Λ)2 + 1 then for every component C of Λ such that (KW + Λ)C > 0 we have r(C) = dim im Res : H 0 (W, KW + Λ) → H 0 (C, (KW + Λ)|C ) ≥ 1. Proof. Without loss of generality we may assume that W does not contain (−1)curves E such that (KW + B)E = 0. We write |KW + Λ| = |H| + G where G is the fixed curve and |H| has at most isolated fixed points; clearly h0 (W, KW + Λ) = h0 (W, H). Our aim is to show that C is not a fixed component of the linear system |KW + Λ|, that is, C is not contained in G. It suffices to show that KW + Λ does not have positive degree on any component of G, which is equivalent to (KW + Λ)G = 0 because KW + Λ is big and nef.

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WENFEI LIU AND SÖNKE ROLLENSKE

Case 1: Suppose that the image of the map ϕ associated to |H| has dimension two. Then both H and KW + Λ are big and nef and thus by assumption (8)

h0 (W, H) = h0 (W, KW + Λ) = (KW + Λ)2 + 1 = (H + G)2 + 1 = H 2 + HG + (H + G)G + 1 ≥ H 2 + 1.

In case of equality we have HG = (H + G)G = 0 which implies HG = G2 = 0 and thus G = 0 by the Hodge-index-theorem and we are done. So assume that h0 (W, H) > H 2 + 1. Then by [TZ92, Lemma 2.1] we have 0 h (W, H) = H 2 + 2. Thus 1 = (KW + Λ)2 − H 2 = 2HG + G2 = HG + (KW + Λ)G. If HG = 0 then G2 ≤ 0 by the Hodge-index-theorem which gives a contradiction. So HG = 1 and (KW + Λ)G = 0 and we are done. Case 2: Suppose that the image of the map ϕ associated to |H| has dimension 1. Our argument follows closely the proof of [Sak80, Thm. 6.1]. Let p : W ∗ → W be a minimal resolution of the linear series |H| so that |p∗ H| = |H ∗ | + E and H ∗ has no base-points. Denoting by A the image of ϕ and by A∗ the Stein factorisation we have a diagram W

ϕ

p

W∗

A

0 (W,K +Λ)−1 W

Ph

s ψ

A∗

Then there is a divisor H of degree n = deg s · deg OA (1) ≥ h0 (W, KW + Λ) − 1 on A∗ such that H ∗ = ψ ∗ H. If F ∗ is a fibre of ψ and F is its image in W then (KW + Λ)2 = n2 F 2 + nF G + (KW + Λ)G ≥ n2 F 2 + nF G ≥ n2 F 2 . Combining all these inequalities we deduce: • If F 2 > 0 then h0 (W, KW + Λ) − 1 ≤ (KW + Λ)2 . Assuming equality we have (KW + Λ)2 = n = F 2 = 1 and F G = (KW + Λ)G = 0. Consequently h0 (W, KW + Λ) = 2 and the fixed part G = 0. • If F 2 = 0 then W = W ∗ and (KW +Λ)2 = 2nF G+G2 . Since (KW +Λ)2 > 0, the fixed part G is non-empty and F G > 0. Thus we get (KW + Λ)2 = nF G + (KW + Λ)G ≥ n ≥ h0 (W, KW + Λ) − 1 Assuming (KW +Λ)2 = h0 (W, KW +Λ)−1 we have (KW +Λ)G = 0, F G = 1 and deg A = h0 (W, KW + Λ) − 1. So in both cases KW + Λ has degree zero on every component of the fixed locus of the linear system |KW + Λ|, which concludes the proof. 4.3. Proof of Theorem 4.1. Using (6) and (7) we work on the minimal semiresolution. If X is irreducible then the proof of the theorem is a simple corollary of the normal case, so the complexity comes from the fact that we glue several components to one surface. More precisely, we have to control how many sections on the normalisation do not descend to the stable log surface. We will proceed by induction on the number k of irreducible components and in each step define a boundary divisor suited to our purpose.

GEOGRAPHY OF GORENSTEIN STABLE LOG SURFACES

15

To set up the induction we order the components of Y¯ such that for 1 ≤ i ≤ k the surface U i := Y1 ∪ · · · ∪ Yi is connected in codimension 1. We define a boundary Λi on U i by the equation ¯ i )| = D ¯ + ∆ ¯ + Z¯j (DU¯ i + Λ Yj Yj Y¯

(j = 1, . . . , i)

j

¯ i = Fi Y¯j → U i . In where DU¯ i is the conductor divisor of the normalisation U j=1 other words, we divide the boundary on Y¯ in a part that is the conductor divisor of U i and the rest. Note that since Y is semi-smooth the surface U i is also a semi-smooth scheme, (U i , Λi ) is a log surface and and (U k , Λk ) = (Y, ∆ + Z). To conclude we show that each of the pairs (U i , Λi ) satisfies pg (U i , Λi ) ≤ (KU i + Λi )2 + 2 and in the case of equality the following four properties hold for each 1 ≤ j ≤ i: (E1 ) The pair (Y¯j , D ¯ + ∆ ¯ + Z¯j ) is a pg -extremal pair (see Proposition 4.6). Yj

Yj

(E2 ) Each irreducible component of U j is smooth, that is, each component of the conductor divisor DU j is contained in two different irreducible components of U j . (E3 ) The linear system |KU j + Λj | defines a morphism whose image is the semilog-canonical model. (E4 ) The intersection Uj−1 ∩ Yj , called the connecting curve for Uj−1 and Yj , is a single smooth rational curve Cj that is mapped isomorphically to a line by the morphism associated to the linear system |KU j + Λj | and deg(KU j + Λj )|C = 1. In particular, Λj 6= 0. j

Base case of the induction: The surface U 1 = Y1 is irreducible and by (7) and Proposition 4.5 we have ¯ 1, D ¯ 1 + Λ ¯ 1 ) ≤ (K ¯ 1 + D ¯ 1 + Λ ¯ 1 )2 + 2 = (KU 1 + Λ1 )2 + 2. pg (U 1 , Λ1 ) ≤ pg (U U U U ¯ 1, D ¯ 1 + If the equality pg (U 1 , Λ1 ) = (KU 1 +Λ1 )2 +2 holds then the normalisation (U U ¯ 1 ) is a pg -extremal pair (see Proposition 4.6); this shows (E1 ). Λ ¯ 1 | and hence the map induced by the Moreover, η ∗ |KU 1 + Λ1 | = |KU¯ 1 + DU¯ 1 + Λ 1 ¯ | on the conductor divisor D ¯ 1 factors through the linear system |KU¯ 1 + DU¯ 1 + Λ U normalisation map and is 2 : 1. By Remark 4.7 any component of the boundary of ¯ 1, D ¯ 1 + Λ ¯ 1 ) is either contracted or embedded by this linear system. Consequently, (U U the conductor is empty and U 1 is normal, which gives (E2 ), and also (E3 ) again by Remark 4.7. The last property (E4 ) is empty for irreducible surfaces, except for the fact that Λ 6= 0, which is true because the conductor is empty. Induction step: Now assume U i−1 satisfies the inequality and conditions (E1 )–(E4 ) in case of equality. Let Ci = U i−1 ∩ Yi be the connecting curve. Note that Ci is a possibly nonconnected smooth curve. By Remark 2.7 (9)

H 0 (U i , KU i + Λi ) = H 0 (U i−1 , KU i−1 + Λi−1 ) ×Ci H 0 (Yi , KYi + Λi |Yi + Ci )

where the right hand side is the vector space fibre product induced by the residue maps rU i−1 and rYi to Ci .

16

WENFEI LIU AND SÖNKE ROLLENSKE

Let rY¯i be the residue map to Ci on the normalisation. Pulling back sections on Yi that vanish along Ci to the normalisation Y¯i is injective, so we can estimate h0 (Yi , KYi + Λi |Y ) − dim im(rYi ) i

¯ i | ) − dim im(r ¯ ) ≤h (Y¯i , KY¯i + DY¯i + Λ Yi Y¯ 0

(10)

i

≤(KY¯i + DY¯i

¯ i | )2 +Λ Y¯

(Lemmata 4.8 and 4.9)

i

=(KYi + Λi |Yi )2 . From equation (9) we get h0 (U i , KU i + Λi ) ≤ h0 (U i−1 , KU i−1 + Λi−1 ) + h0 (Yi , KYi + Λi |Y + Ci ) i

− max{dim im(rUi−1 ), dim im(rYi )} (11)

≤ h0 (U i−1 , KU i−1 + Λi−1 ) + h0 (Yi , KYi + Λi |Y + Ci ) i

− dim im(rYi ) ≤ (KU i + Λi )2 + 2 where we have used the induction hypothesis and (10). It remains to prove that properties (E1 ) to (E4 ) hold for Ui in the case of equality. So assume that h0 (U i , KU i + Λi ) = (KU i + Λi )2 + 2. Then all inequalities in (10) and (11) are equalities, which implies (i) rU i−1 and rYi have the same image; (ii) h0 (Yi , KYi + Λi |Y + Ci ) = (KYi + Λi |Yi + Ci )2 + dim im(rYi ); i (iii) h0 (U i−1 , KU i−1 ) = (KU i−1 + Λi−1 )2 + 2, so U i−1 satisfies (E1 )–(E4 ). By (E3 ), we have dim im(rU i−1 ) ≥ 2 because every component of Ci is embedded by |KU i−1 + Λi−1 |. Thus from (i) and (ii) we get h0 (Yi , KYi + Λi |Y + Ci ) ≥ (KYi + Λi |Yi + Ci )2 + 2, i

so by the base case for the induction equality holds and the pair (Yi , Λi |Yi + Ci ) satisfies properties (E1 )–(E4 ). In particular, U i satisfies (E1 ). Moreover, dim im(rYi ) = 2 and thus by Lemma 4.8 the connecting curve Ci is isomorphic to P1 and KU i + Λi has degree 1 on Ci , which gives the first part of (E4 ) and also that Λi |Y 6= 0 by i the classification of pg -extremal surfaces. Note also that by dimension reasons both rU i−1 and rYi are surjective. By (9) the space |KU i + Λi |∗ is spanned by two natural subspaces A := |KU i−1 + i−1 Λ |∗ and B := |KYi + Λi |Yi + Ci |∗ and their intersection is the line A ∩ B = |(KU i + Λi )|C |∗ . Thus |KU i + Λi ||U i−1 embeds U i−1 into the subspace A by |KU i−1 + Λi−1 | i and |KU i + Λi ||Yi embeds Yi into the subspace B via |KYi + Λi |Yi + Ci | such that Ci is embedded as A ∩ B. In particular, U i−1 and Yi have independent tangent directions along Ci and thus the linear system is an embedding of all of Ui such that Ci is mapped to a line, which proves the second part of (E4 ) and (E3 ). The last part of the proof shows the following: Corollary 4.10 — Let (X, ∆) be a Gorenstein stable log surface such that pg (X, ∆) = (KX + ∆)2 + 2. Let X1 , . . . , Xk be the irreducible components of X. We choose the order such that Vi = X1 ∪ · · · ∪ Xi is connected in codimension 1. Then (i) every irreducible component Xi is a normal stable log surface as in Proposition 4.6,

GEOGRAPHY OF GORENSTEIN STABLE LOG SURFACES

17

(ii) the linear system |KX + ∆| defines an embedding ϕ : X ,→ P = |KX + ∆|∗ , (iii) Ci = Vi ∩ Xi+1 is a smooth irreducible rational curve, (iv ) the linear span of ϕ(Vi ) and of ϕ(Xi+1 ) intersect exactly in the line ϕ(Ci ). In particular, ∆ 6= 0. We believe that these conditions characterise uniquely Gorenstein stable log surfaces on the stable Noether line. 5. Examples 5.1. Surfaces on a string: covering all invariants. We now construct a series of Gorenstein stable surfaces Xk,l with invariants ( k+1 k ≥3 2 KXk,l = k and 1 − k ≤ χ(OXk,l ) = l ≤ k + 2 k = 1, 2 We want to underline the following consequences of these examples 2 (i) The surfaces Xk,1 have P2 = 1 yet KX = k can be arbitrary large. k,1 (ii) For every (a, b) ∈ N × Z such that 1 − a ≤ b ≤ a + 1 or (a, b) ∈ {(1, 3), (2, 4)} 2 = a, χ(O ) = b and with there exists a stable surface X such that KX X normalisation a disjoint union of projective planes. In particular, all possible invariants for minimal surfaces of general type are realised by (non-normal) stable surfaces.

5.1.1. Construction principle. By Theorem 2.3 a stable surface is uniquely deter¯ D, ¯ τ ) consisting of the normalisation, the conductor divisor mined by the triple (X, ¯ preserving the different. To construct and an involution on the normalisation of D Xk,∗ we choose: ¯ = Fk P2 Normalisation: X i=1 ¯ consists of four general lines in each copy of the plane. Conductor: D ¯ is a disjoint union of 4k copies of P1 and each Involution: The normalisation of D ¯ A fixed point free copy contains three marked points that map to nodes of D. involution τ preserving the different is uniquely determined by specifying pairs of lines that are interchanged and the action of τ on the marked points. The important information is contained in the choice of τ . For simplicity, in each copy of P2 we will glue two lines to each other. Different choices for this glueing gives us four different elementary tiles, each containing two lines that still have to be glued. In a second step we choose k of these elementary tiles and specify how to glue them in a circle to get Xk,l . In fact, it will be convenient to work with the minimal semi-resolution f : Yk,l → Xk,l , which is more easily visualised and where some computations are more straightforward. So we blow up all intersection points of the four lines as seen in Figure 3 and e2 and an involution τe which construct a semi-smooth surface Yk,l from k copies of P specifies how to glue the Li in the various components to each other (preserving the intersection points with the exceptional curves). To recover Xk,l from Yk,l we just need to contract all exceptional curves. Alterna¯ k,l , D, ¯ τ ). tively one can simply use the same involution in the triple (X

18

WENFEI LIU AND SÖNKE ROLLENSKE

Figure 3. The basic normal tile. L1

L2 L3

blow up nodes

L4

P2

e2 P

Figure 4. The four different possibilities to glue L3 to L4 up to isomorphism. L1

L2

L1

L2

L34 L34 Type A L1

Type B L2

L34 Type C

L1

L2

L34 Type D

Remark 5.1 — The singularities of the constructed surfaces are very simple to describe (see [KSB88], [Kol12] or [LR12, Sect. 4.2]): apart from smooth and normal crossing points we have only very special degenerate cusps. Assume p is a degenerate cusp on Xk,l . Then its preimage f −1 (p) in the semiresolution Yk,l is a cycle of m f -exceptional curves, which become (−1)-curves in the normalisation. If m = 1 then locally analytically p ∈ Xk,l is isomorphic to the cone over a plane nodal cubic, if m = 2 then locally analytically p ∈ Xk,l is isomorphic to the origin in {x2 + y 2 z 2 = 0} ⊂ C3 (sometimes called T2,∞,∞ ), and if m ≥ 3 then locally analytically p ∈ Xk,l is isomorphic to the cone over a cycle of m independent lines in projective space. By [Ste98, Sect. 3.4] every such surface is locally smoothable, but usually there are global obstructions. e2 , the plane blown up in the intersection 5.1.2. The elementary tiles. Consider P points of four general lines L1 , . . . , L4 . There are six different ways to glue L3 to L4 while preserving the intersections with the exceptional divisor, which up to isomorphism (renaming L1 and L2 ) reduce to four essentially different possibilities. These are given in Figure 4. The surfaces have normal crossing singularities along L34 , the image of L3 and L4 . Note that for esthetic reasons we sometimes use a partly mirrored version of Type A in later figures.

GEOGRAPHY OF GORENSTEIN STABLE LOG SURFACES

19

Figure 5. The surfaces Y1,3 and X1,3 L12

L12 α

α

β resolve deg. cusps γ

β γ

L34 Y1,3

δ β

γ

L34 L12

α X1,3

5.1.3. Warm up: constructing X1,3 . As a starting point we describe in detail the surfaces Y1,3 and X1,3 . We start with one elementary tile of type C. Then an identification of L1 and L2 preserving the intersection points with the exceptional divisor is uniquely determined by the images of the intersection points which we indicate with Greek letters. If we contract all exceptional curves we obtain the stable surface X1,3 . From Figure 5 we read off that on Y1,3 we have four cycles of exceptional curves, two of length one and two of length two. In X1,3 these are contracted to four degenerate cusps. We will 2 confirm below that χ(OX1,3 ) = 3 and KX = 1. 1,3 Either computing the canonical ring directly using [Kol13, Prop. 5.8] or by reverse engeneering one can check that X1,3 is isomorphic to the weighted hypersurface of degree 10 in P(1, 1, 2, 5) with equation z 2 + y(x21 − y)2 (x22 − y)2 . Geometrically, X1,3 is a double cover of the quadric cone in P3 branched over the vertex, a plane section and two double plane sections. From either description we see that X1,3 is the degeneration of a smooth Horikawa surface [BHPV04, VII.(7.1)]. 5.1.4. Computation of invariants. We now explain how to compute the invariants of surfaces constructed as above. We get the self-intersection of the canonical divisor by pulling back to the normalisation: 2 KX = (KX¯ k,l + conductor divisor)2 = k(KP2 + four lines)2 = k. k,l

For the holomorphic Euler characteristic we first compute on the semi-resolution Yk,l . Let DYk,l be the non-normal locus and DY¯k,l be the conductor divisor in the normalisation Y¯k,l . Note that DY¯k,l is the disjoint union of 4k copies of P1 and DYk,l is the disjoint union of 2k copies of P1 . Then χ(OYk,l ) = χ(OY¯k,l ) − χ(ODY¯ ) + χ(ODYk,l ) = kχ(OPe2 ) + (−4k + 2k)χ(OP1 ) = −k. k,l

Let c be the number of degenerate cusps of Xk,l which, by Remark 5.1, corresponds to the number of cycles of exceptional curves in Yk,l . Since by [LR12, Lem. A.7] R1 f∗ OYk,l is a skyscraper sheaf which has length 1 exactly at the degenerate cusps of Xk,l we have by the Leray spectral sequence and the above computation χ(OXk,l ) = χ(OYk,l ) + c = c − k. Going back to X1,3 constructed above, we see that there are exactly four degenerate cusps, so χ(OX1,3 ) = 3 as claimed. Remark 5.2 — It is not very complicated to give a combinatorial formula for the holomorphic Euler characteristic of Xk,l without the use of the semi-resolution and thus avoiding the use of [LR12, Lem. A.7].

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WENFEI LIU AND SÖNKE ROLLENSKE

5.1.5. Construction of Xk,l for 1 − k ≤ l ≤ k + 1. The above computations tell us 2 how to proceed: in order for the surface Xk,l to have KX = k and χ(OXk,l ) = l we k,l glue k copies of the plane in such a way that the resulting surface has exactly k + l degenerate cusps. Alternatively, we construct the semi-resolution Yk,l by glueing k elementary tiles such that there are exactly k + l cycles of exceptional curves. To construct Yk,1−k and thus Xk,1−k we glue k components of type A in a circle as specified in Figure 6. There is only one circle of exceptional curves, thus just 1 = l + k degenerate cusp. Figure 6. The surface Yk,k−1 L12

L12

α β

β γ

γ

α

To get Xk,l for 1 − k < l ≤ 1 we glue 1 − l elementary tiles of type A to k + l − 1 elementary tiles of type B as specified in Figure 7. We read off from the graphical representation that Yk,l contains c = k + l cycles of rational curves and thus Xk,l has k + l degenerate cusps. Figure 7. The surface Yk,l for 1 − k < l ≤ 1 L12

L12 α

α β

β ...

γ

γ

To get Xk,l for 2 ≤ l ≤ k + 1 we glue l − 1 elementary tiles of type D to k − l + 1 elementary tiles of type B as specified in Figure 8. We read off from the graphical representation that Yk,l contains c = k + l cycles of rational curves and thus Xk,l has k + l degenerate cusps also in this case. Figure 8. The surface Yk,l for 2 ≤ l ≤ k + 1 L12

L12 α

α β

β ...

γ

... γ

5.1.6. The surface X2,4 . The last case cannot be constructed by the same strategy as before. But instead, to get X2,4 we just take two copies of (P2 , nodal quartic curve) 2 and let the involution exchange the two curves. The resulting surface has KX =2 2,4 and χ(OX2,4 ) = 4; it is a degeneration of a Horikawa surface.

GEOGRAPHY OF GORENSTEIN STABLE LOG SURFACES

21

5.1.7. Smoothability. Locally all constructed surfaces are smoothable by Remark 5.1. Global smoothability is tricky: we have seen above that X1,3 is smoothable but on the other hand X9,1 cannot be smoothable because minimal surfaces on the BogomolovMiyaoka-Yau line are rigid ball quotients (See also Section 5.3.3). 5.1.8. Further variations. Especially for intermediate values of the invariants there are several other choices of glueing that realise the same invariants. For example, every elementary tile of type A could be replaced by one of type C thereby increasing the number of degenerate cups and hence χ by one. Possibly the resulting surfaces would have different irregularity or geometric genus, but we did not venture into this. 5.1.9. Non-Gorenstein surfaces. If we allow the involution τ to preserve a component of the conductor divisor then it necessarily fixes one of the three marked points. By the classification of slc singularities, the resulting surface is not Gorenstein but has index two. From our building blocks we can also construct non-Gorenstein stable surfaces of index two that violate the stable Noether inequality. To illustrate this we construct 2 a stable surface X3,5 with KX = 3 and χ(OX3,5 ) = 5 given in Figure 9. The two 3,5 lines L1 and L2 are pinched: on each preimage in the involution one of the marked points is fixed and the other two are exchanged. The two fixed points on these lines give pinch points in the semi-smooth surface Y3,5 , which are marked by black dots in the picture. Figure 9. The surface Y3,5 ; the lines L1 and L2 are pinched. L1

L2

5.2. Normal geographical examples. Let C and C 0 be elliptic curves and let S = C × C 0 and fix integers k, l > 0. Pick general points P1 , . . . Pk ∈ C and Q1 , . . . , Ql ∈ C 0 . Let Ci = C × {Qi } i = 1, . . . , k, Cj0 = {Pj } × C 0

j = 1, . . . , l.

Blowing up the k · l points (Pi , Qj ) in S we get a surface Y with (−1)-curves E1,1 , . . . , Ek,l . If Ei (resp. Ej0 ) is the strict transform of Ci (resp. Cj0 ) then Ei and Ej0 are smooth elliptic curves with Ei2 = −l and Ej02 = −k. We construct a surface Xk,l with k + l elliptic singularities by contracting all Ei and Ej0 : Y π

S

σ

Xkl

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WENFEI LIU AND SÖNKE ROLLENSKE

The surface Xk,l exists as an algebraic space. To prove that it is a stable surface it is enough to show that the Cartier divisor KXk,l is ample, for which we use the Nakai–Moishezon criterion ([Kol90, Thm. 3.11]). First note that 2 2 X X X X X 2 Ci + Cj0 ) − Ei,j = kl > 0 KX = KY + Ei + Ej0 = π ∗ ( k,l i

i

j

j

i,j

So let F be an irreducible curve in Xk,l . Its strict transform F¯ ⊂ Y either is one of the π-exceptional curves or is the strict transform of a curve in S. In both cases X X X KX F = π ∗ ( Ci + Cj0 ) − Ei,j F¯ > 0 i

j

i,j

and we are done. To compute χ(OXk,l ) note that 0 = χ(OY ) = χ(OY¯ ) = χ(Rσ∗ OY¯ ) = χ(OXk,l ) − χ(R1 σ∗ OY¯ ) = χ(OXk,l ) − (k + l), because R1 σ∗ OY¯ has length 1 at each elliptic singular point [Rei97, Chapter 4]. Thus we have constructed a normal stable surfaces Xk,l such that 2 χ(OXk,l ) = k + l and KX = kl. k,l 2 In particular, χ(OXk,1 ) = k + 1 = KX + 1, which is the “equality +1” case of the k,l 2 stable Noether inequality, and KXk,k = k 2 > 9χ(OXk,k ) = 18k for k > 18, which confirms that the classical Bogomolov–Miyaoka–Yau inequality does not hold.

5.3. Further examples. 5.3.1. Irregularity. Here we give two examples that show that the irregularity of the normalisation may be larger or smaller than the irregularity of a stable surface. ¯ D) ¯ be a principally polarised abelian Example 5.3 (Drop of irregularity) — Let (X, ¯ surface. Then D is a curve of genus two and thus there is a hyperelliptic involution ¯ τ on D. ¯ D, ¯ τ ) has q(X) = 0 while The stable surface X correponding to the triple (X, ¯ q(X) = 2. Example 5.4 (Increase of irregularity) — In the series of surfaces constructed in ¯ 1,0 ) = Section 5.1 we have χ(OX1,0 ) = 0 so q(X1,0 ) ≥ 1, while on the other hand q(X 2 q(P ) = 0. 5.3.2. Canonical map. Here we note some pathologies of the canonical map that make the classical strategy to prove Noether’s inequality fail for stable surfaces. Example 5.5 (The image of the canonical map need not be equidimensional) — ¯ 1 be a (smooth) del Pezzo surface of degree 1 and D ¯ a nodal curve in | − 2K ¯ |. Let X X1 ¯ 2 with the following properties: D ¯ is contained in X ¯ 2 and Pick a smooth surface X ¯ surjects onto ¯ is very ample and H 1 (X ¯ 2 , ω ¯ ) = 0. In particular H 0 (ω ¯ (D)) ωX¯ 2 (D) X2 X2 0 ¯ H (D, ωD¯ ). ¯ 1 and X ¯ 2 along D. ¯ By Proposition 2.6 We construct a stable surface X by gluing X ¯ 1 , ω ¯ (D)) ¯ = H 0 (X ¯ 1 , ω −1 all sections of H 0 (X ) descend to sections of ωX . So the ¯1 X1 X 1 image of the canonical map restricted to X1 is a P while the image of the canonical map restricted to X2 is a surface.

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Example 5.6 (The image of the canonical map need not be connected) — Let C be a curve of genus at least three which is not hyperelliptic. Glue two copies of C along a point p ∈ C to get a stable curve C 0 . A straightforward computation shows that the canonical map of C 0 has a base-point at p and its image is the disjoint union of two copies of C in the canonical embedding. Now consider the stable surface X = C 0 × C. As a consequence of the above, the base locus of the canonical map coincides with the non-normal locus and the image of the canonical map are two copies of C × C in the canonical embedding. We see that the canonical map is birational while nevertheless its image is not connected. 5.3.3. A family of (fake) fake projective planes. The following example was asked for by Matthias Schütt. It shows explicitly that we should not expect stable surfaces to exhibit a behaviour similar to smooth surfaces with the same invariants. Concretely, 2 = 9 and χ(O ) = 1, the Gieseker moduli space of surfaces of general type with KX X whose elements are usually called fake projective planes, consists of isolated points. We will now construct a 1-dimensional family of stable surfaces with the same invariants thus showing that the number of components of the moduli space of stable surfaces goes up and not all stable surfaces with these invariants are rigid. ¯ =X ¯α t X ¯β t X ¯ γ . Fix in both X ¯α ¯α = X ¯ β = P1 × P1 and X ¯ γ = P2 and X Let X ¯ β the same four horizontal Hx,1 , . . . , Hx,4 and three vertical lines Vx,1 , . . . , Vx,3 and X ¯γ . (x = α, β) and fix four general lines L1 , . . . , L4 ⊂ P2 = X In order to construct a stable surface X we specify an involution τ on G ¯ν = D (Hx,1 t · · · t Hx,4 t Vx,1 t · · · t Vx,3 ) t L1 t · · · t L4 x=α,β

¯α = X ¯ β to identify in the following way: first we use the identity X τ : Vα,i ←→ Vβ,i τ : Hα,i ←→ Hβ,i

(i = 1, 2, 3) (i = 1, 2)

The remaining components all contain 3 marked points for the different and we specify how to glue them by specifying an involution on these points. Points are ¯ in the case denoted by the same symbol if they either map to the same node in X of the Li or if they are identified in the quotient via the gluing of the vertical lines specified above for Hα,i , Hβ,i . Note that the order of the points is important for result of the gluing. τ : Hα,3 = ha, b, ci ←→ h1, 2, 3i = L1 , τ : Hβ,3 = ha, b, ci ←→ h3, 4, 5i = L2 , τ : Hα,4 = hd, e, f i ←→ h2, 5, 6i = L3 , τ : Hβ,4 = hd, e, f i ←→ h1, 6, 4i = L4 . Since (1) is a pushout diagram we see that D has 7 singular points: six arise as images of the nodes of Hα,1 ∪ Hα,2 ∪ Vα,1 ∪ Vα,2 ∪ Vα,3 and the other is the equivalence classes of the point a. The latter has multiplicity 18. Using the normalisation Dν we compute χ(OD ) = 9 − 17 − 6 = −14, χ(OD¯ ) = −5 − 5 − 2 = −12 and consequently, by Proposition 2.5, χ(OX ) = 3 + (−14) − (−12) = 1,

2 ¯ 2 = 9. KX = (KX¯ + D)

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WENFEI LIU AND SÖNKE ROLLENSKE

Note however that we can vary the cross ratio of the four points in P1 corresponding to the 4 horizontal components Hx,i , thus we have a 1-dimensional family of deformations of X. The surface X is locally smoothable but not globally because all smooth fake projective planes are rigid ball quotients [BHPV04].

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Shuichiro Tsunoda and De-Qi Zhang. Noether’s inequality for noncomplete algebraic surfaces of general type. Publ. Res. Inst. Math. Sci., 28(1):21–38, 1992. (cited on p. 10, 12, 14)

Wenfei Liu, Institut für algebraische Geometrie, Gottfried Wilhelm Leibniz Universität Hannover, Welfengarten 1, 30167 Hannover, Germany E-mail address: [email protected] Sönke Rollenske, Fakultät für Mathematik, Universtät Bielefeld, Universitätsstr. 25, 33615 Bielefeld, Germany E-mail address: [email protected]