ULRICH BUNDLES ON CUBIC FOURFOLDS Daniele Faenzi, Yeongrak Kim To cite this version: Daniele Faenzi, Yeongrak Kim. ULRICH BUNDLES ON CUBIC FOURFOLDS. 2020. hal- 03023101v2 HAL Id: hal-03023101 https://hal.archives-ouvertes.fr/hal-03023101v2 Preprint submitted on 25 Nov 2020 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. ULRICH BUNDLES ON CUBIC FOURFOLDS DANIELE FAENZI AND YEONGRAK KIM Abstract. We show the existence of rank 6 Ulrich bundles on a smooth cubic fourfold. First, we construct a simple sheaf E of rank 6 as an elementary modification of an ACM bundle of rank 6 on a smooth cubic fourfold. Such an E appears as an extension of two Lehn-Lehn-Sorger-van Straten sheaves. Then we prove that a general deformation of E(1) becomes Ulrich. In particular, this says that general cubic fourfolds have Ulrich complexity 6. Introduction An Ulrich sheaf on a closed subscheme X of PN of dimension n and degree d is a non-zero coherent sheaf F on X satisfying H∗(X, F(−j)) = 0 for 1 ≤ j ≤ n. In particular, the cohomology table {hi(X, F(j))} of F is a multiple of the cohomology table of Pn. It turns out that the reduced Hilbert polynomial pF (t)= χ(F(t))/ rk(F) of an Ulrich sheaf F must be: n d u(t) := (t + i). n! Yi=1 Ulrich sheaves first appeared in commutative algebra in the 1980s, namely, in the form of maximally generated maximal Cohen-Macaulay modules [Ulr84]. Pioneering work of Eisenbud and Schreyer [ES03] popularized them in algebraic geometry in view of their many connections and applications. Eisenbud and Schreyer asked whether every projective scheme supports an Ulrich sheaf. That this should be the case is now called a conjecture of Eisenbud-Schreyer (see also [ES11]). They also proposed another question about what is the smallest possible rank of an Ulrich sheaf on X. This is called the Ulrich complexity uc(X) of X (cf. [BES17]). Both the Ulrich existence problem and the Ulrich complexity problem have been elucidated only for a few cases. We focus here on the case when X is a hypersurface in Pn+1 over an algebraically closed field k. Using the generalized Clifford algebra, Backelin and Herzog proved in [BH89] that any hypersurface X has an Ulrich sheaf. However, their construction yields an Ulrich sheaf of rank dτ(X)−1, where τ(X) is the Chow rank of X (i.e. the smallest length of an expression of the defining equation of X as sums of products of d linear forms), often much bigger than uc(X). Looking in more detail at the Ulrich complexity problem for smooth hypersurfaces of degree d in Pn+1, the situation is well-understood for arbitrary n only for d = 2. Indeed, in this case the only indecomposable Ulrich bundles on X are spinor bundles, which have rank 2⌊(n−1)/2⌋ [BEH87]. On the other hand, for d ≥ 3, the Ulrich complexity problem is wide open except for a very few small-dimensional cases. For instance, smooth cubic curves and surfaces always X satisfy uc(X) = 1, while for smooth cubic threefolds X we have uc(X) = 2, (cf. [Bea00, Bea02, LMS15]). When X is a smooth cubic fourfold, which is the main object of this paper, there can be several possibilities. In any case, X does not have an Ulrich bundle of rank 1, but some X can have an D.F. partially supported by ISITE-BFC project contract ANR-lS-IDEX-OOOB and EIPHI Graduate School ANR-17-EURE-0002. Y.K. was supported by Project I.6 of the SFB-TRR 195 “Symbolic Tools in Mathematics and their Application” of the German Research Foundation (DFG). Both authors partially supported by F´ed´eration de Recherche Bourgogne Franche-Comt´eMath´ematiques (FR CNRS 2011). 1 2 DANIELE FAENZI AND YEONGRAK KIM Ulrich bundle F of rank 2. Since F is globally generated, one can consider the locus defined by a general global section of F, which is a del Pezzo surface of degree 5. The moduli space of cubic fourfolds containing a del Pezzo surface of degree 5 forms a divisor C14 ⊂ C, so a general cubic fourfold X has uc(X) ≥ 3. A few more cubic fourfolds which have an Ulrich bundle of rank 3 or 4 have been reported very recently by Troung and Yen [TY20]. However, all these cases are special cubic fourfolds which contain a surface not homologous to a complete intersection. Indeed, it turns out that the Ulrich complexity of a very general cubic fourfold is divisible by 3 and at least 6, see [KS20]. On the other hand, it is known that a general cubic fourfold has a rank 9 Ulrich bundle (cf. [IM14, Man19, KS20]). Therefore, the Ulrich complexity of a (very) general cubic fourfold is either 6 or 9. The goal of this paper is to prove the following result. This implies that a general cubic fourfold X satisfies uc(X) = 6. Theorem. Any smooth cubic fourfold admits an Ulrich bundle of rank 6. To achieve this, we use a deformation theoretic argument. Let us sketch briefly the strategy of our proof. As a warm-up it, let us review a construction of a rank-2 Ulrich bundle on a smooth cubic threefold. First, starting from a line L contained in the threefold, one constructs an ACM bundle of rank 2 having (c1, c2) = (0,L). Such a bundle is unstable since it has a unique global section which vanishes along L. By choosing a line L′ disjoint from L, we may take an elementary modification of it so that we have a simple and semistable sheaf E of (c1, c2) = (0, 2L). The sheaf E is not Ulrich, but one can show that its general deformation becomes Ulrich. A similar argument is used to prove the existence of rank 2 Ulrich bundles on K3 surfaces [Fae19] and prime Fano threefolds [BF11]. For fourfolds, twisted cubics play a central role in the construction, rather than lines. Note that twisted cubics in X form a 10-dimensional family. For each twisted cubic C ⊂ X, its linear span V = hCi defines a linear section Y ⊂ X which is a cubic surface. When Y is smooth, the rank-3 sheaf G = ker[3OX → OY (C)] is stable. The family of such stable sheaves of rank 3 forms an 8-dimensional moduli space, which is indeed a very well-studied smooth hyperk¨ahler manifold [LLSvS17, LLMS18]. We will call them Lehn-Lehn-Sorger-van Straten sheaves and the Lehn-Lehn-Sorger-van Straten eightfold. To construct an Ulrich bundle of rank 6, we first consider two Lehn-Lehn-Sorger-van Straten sheaves of rank 3 associated with two points of this manifold. We choose the points so that the associated pair of twisted cubics spans the same smooth surface section of X and intersects at 4 points. Then, we define a simple sheaf E as extension of such sheaves and show that this enjoys a large part of the cohomology vanishing required to be Ulrich. In particular, this will show that E lies in the Kuznetsov category Ku(X) of X, [Kuz04]. Finally, we obtain an Ulrich sheaf by taking a generic deformation F of E in the moduli space of simple sheaves over X and using that the cohomology vanishing of E propagates to F by semicontinuity. This step relies on deformation- obstruction theory of the sheaf as developed in [KM09, BLM+] and makes substantial use of the fact that Ku(X) is a K3 category. The structure of this paper is as follows. In Section 1, we recall basic notions and set up some background. In Section 2, we introduce an ACM bundle of rank 6 which arises as a (higher) syzygy sheaf of a twisted cubic and review some material on Lehn-Lehn-Sorger-van Straten sheaves as syzygy sheaves. Then we take an elementary modification to define a strictly semistable sheaf E of rank 6 whose reduced Hilbert polynomial is u(t). In Section 3, we show that a general deformation of E is Ulrich. We first prove this claim for cubic fourfolds which do not contain surfaces of small degrees other than linear sections. Then we extend it for every smooth cubic fourfold. ULRICH BUNDLES ON CUBIC FOURFOLDS 3 Acknowledgment. We wish to thank F´ed´eration Bourgogne Franche-Comt´eMath´ematiques FR CNRS 2011 for supporting the visit of Y.K. in Dijon. We would like to thank Frank-Olaf Schreyer and Paolo Stellari for valuable advice and helpful discussion. 1. Background Let us collect here some basic material. We work over an algebraically closed field k of charac- teristic other than 2. 1.1. Background definitions and notation. Consider a smooth connected n-dimensional pro- N jective subvariety X ⊆ P and denote by HX the hyperplane divisor on X and OX (1) = OX (HX ). Given a coherent sheaf F on X and t ∈ Z, write F(t) for F ⊗ OX (tHX ). Let F be a torsion-free sheaf on X. The reduced Hilbert polynomial of F is defined as 1 p (t) := χ(F(t)) ∈ Q[t].