Frapporti D.

Gleissner C.

English

In this note we provide a two-dimensional family of smooth minimal three-folds of general type with canonical map of degree 96, improving the previous known bound of 72

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arXiv:1902.03133v2  [math.AG]  11 Nov 2019A FAMILY OF THREEFOLDS OF GENERAL TYPE WITH CANONICALMAP OF HIGH DEGREEDAVIDE FRAPPORTI AND CHRISTIAN GLEISSNERAbstract. In this note we provide a two-dimensional family of smooth minimal threefolds ofgeneral type with canonical map of degree 96, improving the previous known bound of 72.IntroductionIn this paper we consider smooth varieties of general type.In the case of curves it is classically known that the canonical map is either an embedding ora degree 2 map onto P1, the latter happens precisely when the curve is hyperelliptic.In higher dimensions the situation is much less clear and it is natural to ask: “Assuming thatthe canonical map is generically finite, is its degree universally bounded? If so, what is themaximal possible value of the canonical degree?” In this paper by canonical degree we meanthe degree of the canonical map.For surfaces Beauville ([Bea79]) showed that the canonical degree is at most 36, and equalityholds if and only if pg = 3, q = 0, K2 = 36 and the canonical system is base point free.In the threefold case, Hacon [Hac04] established 576 as a bound for the canonical degree.Later on in [DG16] this bound was improved to 360, which can be achieved if and only if pg = 4,q1 = 2, χ(O) = −5, K3 = 360 and the canonical system is base point free. In [Hac04] Hacon alsoexplained that if one allows terminal singularities, there is no bound for the canonical degree,presenting an infinite series of threefolds with index 2 terminal singularities and arbitrarily highcanonical degree.Beauville and Hacon’s proofs rely heavily on the Miyaoka-Yau inequality, which cannot beused to control the canonical degree in higher dimensions.Date: November 12, 2019.2000 Mathematics Subject Classification. 14J30, 14L30, 14.01.Key words and phrases. Threefolds of general type, canonical map, finite group actions.The authors thank S. Coughlan for inspiring conversations and a careful reading of the paper. A very specialthanks goes to R. Pignatelli for the inspiring conversation at the Conference “A Journey through Projectiveand Birational Geometry Together with Marco Andreatta” (Trento, January 2019), where he encouraged andhelped the authors to expose the construction using an elementary approach.The present work took place in the framework of the ERC Advanced grant n. 340258-TADMICAMT.The first author is member of G.N.S.A.G.A. of I.N.d.A.M..12 D. FRAPPORTI, C. GLEISSNERFor surfaces, as well as for Gorenstein minimal threefolds, the maximal value can be achievedonly if the Miyaoka-Yau inequality becomes an equality, i.e. for ball-quotients. Unfortunatelythese are notoriously hard to handle and it is not clear if the bounds are sharp. In [Yeu17]Yeung claims the existence of a surface realizing degree 36, but the proof seems to have a gap,as pointed out by the Mathscinet Reviewer (MR Number: MR3673651). More recently Rito[Rit19] used the Borisov-Keum equations of a fake projective plane to costruct a surface havingcanonical map of degree 36.Leaving ball-quotients aside, one can still try to construct examples having high canonicaldegree. For surfaces, the highest canonical degrees obtained so far are: 16 due to Persson([Per78], double covering of a Campedelli surface), 24 due to Rito ([Rit17], abelian covering ofa blow-up of P2), and 32 due to the second author together with Pignatelli and Rito ([GPR18],product-quotient surface). In dimension 3 the situation is less established. The highest knownvalues of the canonical degree are 32, 48, 64 and 72, which can be achieved simply by taking theproduct C × S, where C is a hyperelliptic curve and S is one of the above mentioned surfaces.Other examples with canonical degree 32 and 64 were constructed in [Cai08].In this paper we set a new record for canonical degree of a smooth threefold:Theorem (Theorem 2.3). There exists a two dimensional family of smooth threefolds X withcanonical degree 96 and whose canonical image is a quadric.The family lives in a 6-dimensional family of threefolds isogenous to a product. Threefoldsisogenous to a product are a special case of product-quotient varieties (see [Cat00, FG16, Gle19]),i.e. varieties birational to the quotient of the product of smooth projective curves by the actionof a finite group.In recent years the intensive work on product-quotient varieties has produced several inter-esting examples, e.g. the mentioned example of a surface of general type with canonical map ofdegree 32; new topological types for surface of general type, in particular a family of surfacesof general type with K2 = 7, pg = q = 2 (see [CF18] and the references therein); and recentlythe first examples of rigid but not infinitesimally rigid compact complex manifolds ([BP18]).For other interesting applications see [Cat17], [FGP18], [GRVR18], [LP16] and [LP18].We point out that we originally found this 2-dimensional family using the technology de-veloped in the papers mentioned above, together with the theory of abelian coverings (see[Lie03, Par91]). However, we decided not to use the language of product-quotients here, but togive a simple and self-contained description using explicit equations instead.The paper is organized as follows: in Section 1 we construct the 6-dimensional family and inSection 2 we prove the main Theorem.A FAMILY OF THREEFOLDS OF GENERAL TYPE WITH CANONICAL MAP OF HIGH DEGREE 31. The constructionLet Ca,b be the curve in P4 defined by the equations(1.1) x22 = x20 − x21 , x23 = x20 − ax21 , x24 = x20 − bx21 ,where a, b ∈ C \ {0, 1}, a 6= b. The curve Ca,b is then a smooth canonical curve of genus 5. Wenote that Ca,b is invariant under the (Z2)4-action on P4 given byei : xi 7→ −xi xj 7→ xj (i 6= j)where {e1, . . . , e4} is the standard basis of (Z2)4 and e0 := e1 + e2 + e3 + e4.The map π : Ca,b → P1, π(x0 : x1 : x2 : x3 : x4) = (x20 : x21) is a covering of the projective linebranched in 5 points:p0 = [0 : 1] , p1 = [1 : 0] , p2 = [1 : 1] , p3 = [a : 1] , and p4 = [b : 1] .Since this map has degree 16 and is (Z2)4-equivariant, it is the quotient map, and the stabilizerof a ramification point in π−1(pi) is the group of order 2 generated by ei.We consider now the threefold T := Ca1,b1 ×Ca2,b2 ×Ca3,b3 ⊂ P4x×P4y×P4zwith the “twisted”(Z2)4 action defined byei(x,y, z) := (ei · x, (Aei) · y, (A2ei) · z) ,where x := (x0 : x1 : x2 : x3 : x4) (similarly for y and z) and A ∈ GL(4,Z2) is the matrixA :=0 1 0 10 1 1 11 1 1 01 0 1 0and A2 =1 1 0 10 0 1 11 1 0 01 0 1 1.Remark 1.1. 1) A is chosen so that{e0, . . . , e4, Ae0, . . . , Ae4, A2e0, . . . , A2e4} = (Z2)4 \ {0} .2) A has order 3 and satisfies I + A + A2 = 0.The rest of the paper is devoted to the study of the canonical map of the quotient threefold(1.2) X := (Ca1,b1 × Ca2,b2 × Ca3,b3)/(Z2)4 .4 D. FRAPPORTI, C. GLEISSNERLemma 1.2. The threefold X is smooth of general type, with ample canonical class KX andinvariantsχ(OX) = −4 , K3X = −48χ(OX) = 192 .Proof. The ei are the unique non-trivial elements of (Z2)4 having fixed points on the curve.According to Remark 1.1, Aej 6= ek for all j, k, therefore the (Z2)4-action on the product T isfree and X is smooth.Since T is of general type, with ample canonical class and the action is free, the quotient Xis also of general type with ample canonical class. Moreover, by the freeness of the action itfollows16χ(OX) = χ(OT ) =3∏i=1(1− g(Cai,bi)) = −4316K3X = K3T = 63∏i=1(2g(Cai,bi)− 2) = −48χ(OT )Remark 1.3. Since Cai,bi is a canonical curve, the restrictionH0(P4,O(1)) −→ H0(Cai,bi, KCai,bi )is an isomorphism. ThereforeH0(Ca1,b1 , KCa1,b1 ) = 〈x0, x1, x2, x3, x4〉 ,H0(Ca2,b2 , KCa2,b3 ) = 〈y0, y1, y2, y3, y4〉 ,H0(Ca3,b3 , KCa2,b3 ) = 〈z0, z1, z2, z3, z4〉 .Lemma 1.4. The canonical system of X is spanned bysi := xiyizi , i = 0, . . . , 4 .In particular pg(X) = 5.Proof. By Künneth’s formula we have the following isomorphismH0(KT ) ∼= H0(KCa1,b1 )⊗H0(KCa2,b2 )⊗H0(KCa3,b3 ) = 〈xiyjzk〉i,j,k.The (Z2)4-action on the product induces an action on H0(KT ) via pull-back:e∗α(xiyjzk) = e∗α(xi) · (Aeα)∗(yj) · (A2eα)∗(zk) = (−1)nα(i,j,k)xiyjzk ,A FAMILY OF THREEFOLDS OF GENERAL TYPE WITH CANONICAL MAP OF HIGH DEGREE 5where nα(i, j, k) = Iiα +Ajα +A2kα for α = 1, 2, 3, 4; and I0α := 0, A0α := 0, A20α := 0 for all α.By the freeness of the action it holdsH0(X,KX) = H0(T,KT )(Z2)4 = 〈xiyjzk | nα(i, j, k) = 0 , ∀α〉 ,and we concludeH0(X,KX) = 〈x0y0z0, x1y1z1, x2y2z2, x3y3z3, x4y4z4〉 ,since Iiα + Ajα + A2kα = 0 for all α if and only if i = j = k (cf. Remark 1.1). Remark 1.5. In a similar way one can determine all the other Hodge numbers of X :q2(X) = 0 , q1(X) = 0 , h1,1(X) = 3 , h2,1(X) = 15 .2. The Canonical MapTheorem 2.1. The canonical system |KX | is base point free, and the canonical image is ahypersurface in P4.Proof. It follows from (1.1) and the conditions on the parameters ak and bk that xi, xj (resp.yi, yj and zi, zj) with i 6= j cannot vanish simultaneously on Ca1,b1 (resp. Ca2,b2 and Ca3,b3).Hence the sections si = xiyizi have no common zeros and the canonical system |KX | is basepoint free.Since KX is ample, the image of the canonical map is a threefold, otherwise there would exista curve C ⊂ X with KX .C = 0. Note that the construction of X depends on 6 parameters a1, b1, a2, b2, a3, b3. We now showthat there is a 2-dimensional subfamily whose elements have canonical degree 96. Since(degϕKX) · deg(ϕKX(X)) = K3X = 192 = 2 · 96 ,and the ϕKX (X) is non-degenerate, we observe that 96 is the maximal possible canonical degree.This maximum is achieved if and only if the image is a quadric in P4, i.e. the sections si satisfya quadratic relation.Proposition 2.2. The sections si satisfy a non-trivial quadratic relation of the form(2.1) λ0s20 + λ1s21 + λ2s22 + λ3s23 + λ4s24 = 0if and only if a1 = a2 = a3 and b1 = b2 = b3.6 D. FRAPPORTI, C. GLEISSNERProof. Using the equations of the curves (1.1), the existence of a non-trivial quadratic relationas in (2.1) is equivalent to vanishing of the following expressionλ0x20y20z20 + λ1x21y21z21 + λ2(x20 − x21)(y20 − y21)(z20 − z21)+λ3(x20 − a1x21)(y20 − a2y21)(z20 − a3z21) + λ4(x20 − b1x21)(y20 − b2y21)(z20 − b3z21)for some 0 6= (λ0, . . . , λ4) ∈ C5.In other words, there exists a non-trivial quadratic relation as in (2.1) if and only if thefollowing system of linear equations has a non-trivial solution:1 0 1 1 10 1 −1 −a1a2a3 −b1b2b30 0 1 a1 b10 0 1 a2 b20 0 1 a3 b30 0 1 a1a2 b1b20 0 1 a2a3 b2b30 0 1 a1a3 b1b3·λ0λ1λ2λ3λ4= 0 .This means that the matrix has rank at most 4. Using elementary matrix transformationsunder using the conditions ai, bi /∈ {0, 1} and ai 6= bi, the matrix becomes1 0 1 1 10 1 −1 −a1a2a3 −b1b2b30 0 1 a1 b10 0 0 1b1(b2 − 1)a1(a2 − 1)0 0 0 0 (b3 − 1)(a2 − 1)− (a3 − 1)(b2 − 1)0 0 0 0 (b2 − b1)a1(a2 − 1)− (a2 − a1)b1(b2 − 1)0 0 0 0 (b3 − b1)a1(a2 − 1)− (a3 − a1)b1(b2 − 1)0 0 0 0 (b2b3 − b1)a1(a2 − 1)− (a2a3 − a1)b1(b2 − 1)which has rank 4 if and only if the following equations hold:A FAMILY OF THREEFOLDS OF GENERAL TYPE WITH CANONICAL MAP OF HIGH DEGREE 7(2.2)i) (b3 − 1)(a2 − 1) = (a3 − 1)(b2 − 1)ii) (b2 − b1)a1(a2 − 1) = (a2 − a1)b1(b2 − 1)iii) (b3 − b1)a1(a2 − 1) = (a3 − a1)b1(b2 − 1)iv) (b2b3 − b1)a1(a2 − 1) = (a2a3 − a1)b1(b2 − 1)We claim that (2.2) is equivalent to(2.3) a1 = a2 = a3 and b1 = b2 = b3 ,under the conditions on the ai’s and bi’s. Clearly, it suffices to prove that a solution of (2.2) isalso a solution of (2.3). We distinguish two cases:Case b1 = b2. Equation ii) implies a1 = a2. We resolve i) after a3 and substitute it in iii):(b3 − b2)a2(a2 − 1) =(a2 − 1)(b3 − b2)(b2 − 1)b2(b2 − 1).The factors (a2 − 1) and (b2 − 1) cancel out and the equation becomes(b3 − b2)(a2 − b2) = 0.Since a2 6= b2, we conclude b2 = b3. This implies a1 = a3 using iii) again.Case b1 6= b2. By ii) we have a1 6= a2 and we are allowed to divide iv) and iii) by ii):b2b3 − b1b2 − b1=a2a3 − a1a2 − a1andb3 − b1b2 − b1=a3 − a1a2 − a1.Furthermore we can assume that b3 6= b1, since b3 = b1 would imply b1 = b2 = b3 similarly tothe previous case. We rewrite the fractions above asb2b3(a2 − a1)− b1(a2 − a1) = a2a3(b2 − b1)− a1(b2 − b1)andb3(a2 − a1)− b1(a2 − a1) = a3(b2 − b1)− a1(b2 − b1)and subtract them:(2.4) (b2b3 − b3)(a2 − a1) = (a2a3 − a3)(b2 − b1).Next we resolve ii) after (a2 − a1) and substitute in (2.4). This simplifies to b3a1 = a3b1. Weuse this equation to rewrite iii) as(b3 − b1)a1(a2 − b2) = 0.8 D. FRAPPORTI, C. GLEISSNERSince a2 6= 0 and a2 6= b2, we conclude that b1 = b3, a contradiction. As consequence we get:Theorem 2.3. There exists a two dimensional family of smooth threefolds X of general typewith canonical degree 96 and whose canonical image is a quadric.References[Bea79] A. Beauville. L’application canonique pour les surfaces de type général. Invent. Math., 55(2):121–140,1979.[BP18] I. Bauer and R. Pignatelli. Rigid but not infinitesimally rigid compact complex manifolds. arXive-prints: 1805.02559, May 2018.[Cai08] J.-X. Cai. Degree of the canonical map of a Gorenstein 3-fold of general type. Proc. Amer. Math.Soc., 136(5):1565–1574, 2008.[Cat00] F. Catanese. Fibred surfaces, varieties isogenous to a product and related moduli spaces. AmericanJournal of Mathematics, 122(1):1–44, 2000.[Cat17] F. Catanese. Canonical surfaces of higher degree. Rend. Circ. Mat. Palermo (2), 66(1):43–51, 2017.[CF18] N. Cancian and D. Frapporti. On semi-isogenous mixed surfaces. Math. Nachr., 291(2-3):264–283,2018.[DG16] R. Du and Y. Gao. On the canonical degrees of Gorenstein threefolds of general type.Geom. Dedicata,185:123–130, 2016.[FG16] D. Frapporti and C. Gleißner. On threefolds isogenous to a product of curves. Journal of Algebra,465:170–189, 2016.[FGP18] F. Favale, C. Gleissner, and R. Pignatelli. The pluricanonical systems of a product-quotient variety.To appear on: Galois Covers, Grothendieck-Teichmüller Theory and Dessins d’Enfants - Interactionsbetween Geometry, Topology, Number Theory and Algebra, Frank Neumann, Sibylle Schroll (ed.),Springer Proceedings in Mathematics & Statistics , 2018.[Gle19] C. Gleissner. Mixed threefolds isogenous to a product. J. Algebra Appl., 18(12):1950239, 28, 2019.[GPR18] C. Gleissner, R. Pignatelli, and C. Rito. New surfaces with canonical map of high degree. arXive-prints: 1807.11854, July 2018.[GRVR18] J. Grivaux, J. Restrepo Velasquez, and E. Rousseau. On Lang’s conjecture for some product-quotientsurfaces. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 18(4):1483–1501, 2018.[Hac04] C. D. Hacon. On the degree of the canonical maps of 3-folds. Proc. Japan Acad. Ser. A Math. Sci.,80(8):166–167, 2004.[Lie03] C. Liedtke. Singular abelian covers of algebraic surfaces. Manuscripta Math., 112(3):375–390, 2003.[LP16] M. Lönne and M. Penegini. On asymptotic bounds for the number of irreducible components of themoduli space of surfaces of general type II. Doc. Math., 21:197–204, 2016.[LP18] M. Lönne and M. Penegini. On Zariski Multiplets of Branch Curves from Surfaces Isogenous to aProduct. To appear on: Michigan Mathematical Journal, 2018.[Par91] R. Pardini. Abelian covers of algebraic varieties. J. Reine Angew. Math., 417:191–213, 1991.A FAMILY OF THREEFOLDS OF GENERAL TYPE WITH CANONICAL MAP OF HIGH DEGREE 9[Per78] U. Persson. Double coverings and surfaces of general type. In Algebraic geometry (Proc. Sympos.,Univ. Tromsø, Tromsø, 1977), volume 687 of Lecture Notes in Math., pages 168–195. Springer,Berlin, 1978.[Rit17] C. Rito. A surface with canonical map of degree 24. Internat. J. Math., 28(6):1750041, 10, 2017.[Rit19] C. Rito. Surfaces with canonical map of maximum degree. arXiv e-prints: 1903.03017, March 2019.[Yeu17] S.-K. Yeung. A surface of maximal canonical degree. Math. Ann., 368(3-4):1171–1189, 2017.Davide Frapporti, Christian Gleißner;University of Bayreuth, Lehrstuhl Mathematik VIII;Universitätsstraße 30, D-95447 Bayreuth, Germany

A family of threefolds of general type with canonical map of high degree

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A family of threefolds of general type with canonical map of high degree

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