ON THE DEGREE OF THE CANONICAL MAP OF A SURFACE OF GENERAL TYPE MARGARIDA MENDES LOPES AND RITA PARDINI A Ciro, maestro e amico. Abstract. Let X be a minimal complex surface of general type such that its image Σ via the canonical map ' is a surface; we denote by d the degree of '. In this expository work, first of all we recall the known possibilities for Σ and d when ' is not birational, which are quite a few, and then we consider the question of producing concrete examples for all of them. We present the two main methods of construction of such examples and we give several instances of their application. We end the paper outlining the state of the art on this topic and raising several questions. 2020 Mathematics Subject Classification: 14J29. Keywords: surface of general type, canonical map, canonical degree Contents 1. Introduction 1 2. The canonical map 2 3. Two constructions 4 4. Examples 12 5. Remarks and open questions 18 References 20 1. Introduction arXiv:2103.01912v1 [math.AG] 2 Mar 2021 In his epochal paper [Be79] Beauville undertook the study of the canonical map of surfaces of general type, bringing to light the great variety of possible behaviours of this map. In spite of some later refinements (see for instance [Xi86], [Xi85]) and of the great many examples in the literature (too many to be listed here), several questions still remain open. Research partially supported by FCT/Portugal through UID/MAT/04459/2020 and by project PRIN 2017SSNZAW 004 \Moduli Theory and Birational Classification” of Italian MIUR. The first author is a member of Centro de An´aliseMatem´atica,Geometria e Sistemas Din^amicosof T´ecnico/Universidade de Lisboa. The second author is a member of GNSAGA of INDAM. 1 2 MARGARIDA MENDES LOPES AND RITA PARDINI In this paper we focus on the case when the canonical image is a surface and the canonical map is not birational, with the aim to offer the reader a quick guide to the topic. In section 2 we summarize the known general results and in section 3 we describe the two main methods of construction of examples found in the literature, that is, generating pairs and abelian covers. In section 4 we apply the methods of section 3 to construct an assortment of examples, with the aim of giving a taste of how the methods work. In section 5 we collect some final remarks and several open questions. We have tried to keep the exposition reader's friendly, hoping that the paper may serve as an introduction to this fascinating topic, besides being a useful reference for surface experts. Notations and Conventions: We work over the complex numbers. All varieties are assumed to be projective and irreducible, unless otherwise spec- ified. For a product variety X1×X2 we denote by pi : X1×X2 ! Xi, i = 1; 2, the two projections and, given Li 2 Pic(Xi), we denote by L1 L2 the line ∗ ∗ bundle p1L1 ⊗ p2L2 on X1 × X2. We denote linear equivalence of divisors by ≡. 0 2 For a smooth surface X we write as usual pg(X) := h (KX ) = h (OX ) 0 1 1 and q(X) := h (ΩX ) = h (OX ); in case X is singular we set pg(X) = pg(Y ) and q(X) = q(Y ) for Y any desingularization of X. As usual an effective divisor on a surface is said to be normal crossings if its only singularities are ordinary double points and it is said to be simple normal crossings if it is normal crossings and in addition all its irreducible components are smooth. 2. The canonical map In this section, we denote by X a surface of general type with pg := pg−1 pg(X) ≥ 3 and by ': X ! P the canonical map; we assume that the image Σ of ' is a surface and we set d := deg '. 2.1. The canonical image. The main result on the canonical image in case it is a surface is due to Beauville and is the following: Theorem 2.1 ([Be79], Thm. 3.1). There are the following possibilities: (A) pg(Σ) = 0; (B) pg(Σ) = pg(X) and Σ is a canonical surface, namely it is the canon- ical image of a surface of general type Z with birational canonical map. As explained in [Be79] (see Example 4.7) for any surface T with pg = 0 one can find minimal surfaces of general type X such that the image of the canonical map of X is T , by taking a suitable Z2-cover. On the other hand, surfaces in case (B) with d ≥ 2 were thought for a long time not to exist. The first example of such a phenomenon is a surface with d = 2 whose 3 canonical image is a quintic surface in P with 20 double points (see [Be79], [Ca81]). ON THE DEGREE OF THE CANONICAL MAP OF A SURFACE OF GENERAL TYPE3 2.2. Bounds on the canonical degree. Here we look at the bounds on the degree d of the canonical map. Some of the examples referred to below can be found in §4. Recall that writing KX ≡ M+Z, where Z is the fixed part of the canonical linear system jKX j, one has M 2 ≥ d deg Σ: Minimal surfaces S of general type with birational canonical map satisfy 2 KS ≥ 3pg(S) + q(S) − 7 (see [De82, Thm. 3.2]). So in case (B) one has deg Σ ≥ 3pg(X) + q(Σ) − 7 whilst in case (A) one has deg Σ ≥ pg(X) − 2, p −1 since Σ generates P g . 2 2 Assume now that X is minimal. Then one has KX ≥ M and so in case (A), 2 KX ≥ d(pg(X) − 2) and in case (B) 2 KX ≥ d(3pg(X) + q(Σ) − 7): Combining the above inequalities with the Bogomolov-Miyaoka-Yau bound 2 KX ≤ 9χ(OX ) one obtains for case (A) (2.1) 27 − 9q(X) ≥ (d − 9)(pg(X) − 2) and for case (B) (2.2) 30 − 9q(X) − dq(Σ) ≥ (d − 3)(3pg(X) − 7) Thus, as already pointed out in [Be79], in case (A) d ≤ 9 if pg ≥ 30 and in case (B) d ≤ 3 if pg(X) ≥ 13. Later on Xiao Gang in [Xi86] showed that in fact in case (A) if pg(X) > 132 then deg ' ≤ 8. For degree 8 there exist examples with unbounded pg as shown by Beauville (see Example 4.8). More recently in [Ng20] Nguyen Bin produced several other such examples. Surfaces in case (B) with d ≥ 2 are much more difficult to find. Note that minimal surfaces in case (B) with d = 2 must satisfy 2 (2.3) KX ≥ 6pg(X) + 2q(Σ) − 14 whilst if d = 3 they must satisfy 2 (2.4) KX ≥ 9pg(X) + 3q(Σ) − 21 and as such are very near the Bogomolov-Miyaoka-Yau bound. Families with unbounded pg and d = 2 are known (see §4) but for d ≥ 3 one knows only sporadic examples. Many other consequences of the above formulas can be drawn. To point a few: (1) d ≤ 9 whenever q(X) ≥ 3; (2) as noted first in [Pe77], the maximum possible degree is 36 and can be attained only if pg = 3, q = 0; (3) if q(X) > 0 the maximum possible degree is 27 and can be attained only if pg = 3, q = 1; 4 MARGARIDA MENDES LOPES AND RITA PARDINI (4) in case (B) the maximum possible degree is 9 and can be attained only if pg = 4, q = 0; (5) in case (B) if d = 3, q(X) ≤ 3. C. Rito in [Ri19] uses the Borisov-Keum equations of a fake projective plane and the Borisov-Yeung equations of the Cartwright-Steger surface (the 2 unique known minimal surface of general type with K = 9; pg = q = 1) to show the existence of a surface with pg = 3; q = 0 and canonical map of degree 36, and of a surface with pg = 3; q = 1 and canonical map of degree 2 27 (as in (2) and (3) above). The first surface is an ´etale Z2-cover of the fake projective plane and the second an ´etale Z3-cover of the Cartwright-Steger surface. Let us also remark that (again as explained in [Be79]) that if pg ≥ 25 and Σ has Kodaira dimension ≥ 0 then d ≤ 4. This is an immediate consequence n of the fact that a surface in P of degree less than 2n − 2 has Kodaira dimension −∞ and if the degree is 2n − 2 either it has Kodaira dimension −∞ or it is a K3-surface that, having pg = 1, cannot be the image of a canonical map by Theorem 2.1. 3. Two constructions We describe here the two main constructions used in the literature to produce examples of surfaces with non birational canonical map. 3.1. Generating pairs. The first instance of this construction is due to Beauville (cf. [Ca87], §2.9), who used it to produce the first known un- bounded sequence of surfaces falling in case (B) of Theorem 2.1. Later the construction was studied systematically in [CPT00] and [CPT03], producing more unbounded sequences of such examples. The idea is to start with a finite map h: V ! W of surfaces and a line bundle L on W satifying certain conditions (cf. Definition 3.1 below) and use it to construct a sequence of surfaces Xn, n ≥ 3 such that the canonical image Σn of Xn is a surface and: • pg(Σn) = 0 and the canonical map of Xn has degree twice the degree of h if the general curve C 2 jLj is hyperelliptic; • pg(Σn) = pg(Xn) and the canonical map of Xn has degree equal to the degree of h if the general curve C 2 jLj is not hyperelliptic; • limn!1 pg(Xn) = +1 More precisely, we formulate the following: Definition 3.1.