A. We show that any polarized K3 surface supports special Ulrich bundles of rank 2.Given an n-dimensional closed subvariety X ⊂ P N , a coherent sheaf F on X is Ulrich if H * (F (−t)) = 0 for n consecutive values of t. We refer to [Cos17,Bea18] for an introduction. We mention that Ulrich sheaves are related to Chow forms (this was the main motivation for they study in [ESW03]), to determinantal representations and generalized Clifford algebras, to Boij-Söderberg theory (cf. [SE10]) to the minimal resolution conjecture, and to the representation type of varieties (cf. [FP15]).Conjecturally, Ulrich sheaves exist for any X , see [ESW03]. ey are known to exist for several classes of varieties e.g. complete intersections, curves, Veronese, Segre, Grassmann varieties. Low-rank Ulrich bundles on surfaces have been studied intensively, and Ulrich bundles of rank 2 (or sometimes 1) are known in many cases. We refer to [Cas17,Bea18] for a survey and further references. Let us only review some of the cases that are most relevant for us, namely among surfaces with trivial canonical bundle.In [Bea16], Ulrich bundles of rank 2 are proved to exist on abelian surfaces. In [AFO17], it is proved that K3 surfaces support Ulrich bundles of rank 2, provided that some Noether-Lefschetz open condition is satisfied. e case of quartic surfaces was previously analyzed in detail in [CKM12]. e main techniques used so far are the Serre construction starting from points on X and Lazarsfeld-Mukai bundles.In this note, we show that any K3 surface supports an Ulrich bundle E of rank 2 with c 1 (E) = 3H , for any polarization H . So these bundles are special, cf. [ESW03]. We allow singular surfaces with trivial canonical bundle. e main tool is a sort of enhancement of Serre's construction based on unobstructedness of simple sheaves on a K3 surface.Let us state the result more precisely. We work over an algebraically closed field k. Let X be an integral (i.e. reduced and irreducible) projective surface with ω X ≃ O X and H 1 (O X ) = 0. We denote by X sm the smooth locus of X .Fix a very ample divisor H on X . Under the closed embedding given by the complete linear series |O X (H )| we may view X as a subvariety of some projective space P . A hyperplane section C of X is a projective curve of arithmetic genus with ω C ≃ O C (H ), where H also denotes the restriction of H to C. We may choose C to be integral too.A locally Cohen-Macaulay sheaf E on X is arithmetically Cohen-Macaulay (ACM) if H 1 (E(tH )) = 0 for all t ∈ Z. A special class of ACM sheaves are Ulrich sheaves, which are characterized by the property H * (E(−tH )) = 0 for a pair of consecutive integers t. Sometimes these integers are required to be 1 and 2 in which case we speak of an initialized Ulrich sheaf. Of course all these notions depend on the polarization H . We call simple a sheaf whose only endomorphisms are homotheties. eorem 1. Let X and H be as above. en there exists a simple Ulrich vector bundle of rank 2 on X whose determinant is O X (3H ).
