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**Ulrich bundles on \(K3\) surfaces.**
*(English)*
Zbl 1436.14081

Let \(X \subset \mathbb{P}^N\) be an \(n\)-dimensional projective variety. A coherent sheaf \(\mathcal F\) on \(X\) is Ulrich if \(H^{*}(X, \mathcal F(-t)) = 0\) for \(1 \le t \le n\). Ulrich sheaves have several applications, including determinantal representations of Chow forms, Clifford and Roby-Clifford algebras, the minimal resolution conjecture, Boij-Söderberg theory, etc. Eisenbud and Schreyer asked whether any given \(X\) supports an Ulrich sheaf, whose positive answer is now called as their conjecture [D. Eisenbud and F.-O.Schreyer, J. Am. Math. Soc. 16, No. 3, 537–575, Appendix 576–579 (2003; Zbl 1069.14019); Abel Symp. 6, 35–48 (2011; Zbl 1248.14058)].

Still the conjecture is open for algebraic surfaces, however, a number of cases were known to have locally free Ulrich sheaves of rank \(\le 2\). In particular, Ulrich bundles of rank \(2\) are proved to exist on minimal surfaces of Kodaira dimension \(0\), possibly except for \(K3\) surfaces [A. Beauville, Eur. J. Math. 4, No. 1, 26–36 (2018; Zbl 1390.14130); G. Casnati, Int. J. Math. 28, No. 8, Article ID 1750061, 18 p. (2017; Zbl 1435.14042)]. Ulrich bundles of rank \(2\) on \(K3\) surfaces were also known to exist for quartic surfaces [E. Coskun et al., Doc. Math. 17, 1003–1028 (2012; Zbl 1274.14050)], and for \(K3\) surfaces satisfying some Brill-Noether conditions [M. Aprodu et al., J. Reine Angew. Math. 730, 225–249 (2017; Zbl 1387.14088)]. In particular, \(K3\) surfaces having Picard number \(1\) has an Ulrich bundle of rank \(2\) of determinant \(\mathcal O_S(3)\) (such an Ulrich bundle is called special).

The paper under review completes the existence of special Ulrich bundles for every polarized \(K3\) surface. Let \(X\) be an integral projective surface with \(\omega_X \cong \mathcal O_X\) and \(H^1(\mathcal O_X)=0\), and let \(H = \mathcal O_X(1)\) be a very ample divisor on \(X\) which defines an embedding \(X \hookrightarrow \mathbb{P}^g\). Using Serre’s corrsepondence and an elementary modification with respect to a point, the author constructs a simple sheaf \(\mathcal E^{\eta}\) with \(c_1 (\mathcal E^{\eta}) = H\), \(c_2 (\mathcal E^{\eta}) = g+3\), \(H^{*} (\mathcal E^{\eta}) = 0\) (Lemma 2–4). Even though \(\mathcal E^{\eta}\) itself is not an Ulrich bundle, its general deformation becomes an Ulrich bundle (Lemma 5). The author also describes the moduli space of stable Ulrich bundles of any even rank on \(X\) (Theorem 7). Note that this construction is similar to construction of ACM/Ulrich bundles on Fano threefolds [M. C. Brambilla and D. Faenzi, Mich. Math. J. 60, No. 1, 113–148 (2011; Zbl 1223.14047)]

Still the conjecture is open for algebraic surfaces, however, a number of cases were known to have locally free Ulrich sheaves of rank \(\le 2\). In particular, Ulrich bundles of rank \(2\) are proved to exist on minimal surfaces of Kodaira dimension \(0\), possibly except for \(K3\) surfaces [A. Beauville, Eur. J. Math. 4, No. 1, 26–36 (2018; Zbl 1390.14130); G. Casnati, Int. J. Math. 28, No. 8, Article ID 1750061, 18 p. (2017; Zbl 1435.14042)]. Ulrich bundles of rank \(2\) on \(K3\) surfaces were also known to exist for quartic surfaces [E. Coskun et al., Doc. Math. 17, 1003–1028 (2012; Zbl 1274.14050)], and for \(K3\) surfaces satisfying some Brill-Noether conditions [M. Aprodu et al., J. Reine Angew. Math. 730, 225–249 (2017; Zbl 1387.14088)]. In particular, \(K3\) surfaces having Picard number \(1\) has an Ulrich bundle of rank \(2\) of determinant \(\mathcal O_S(3)\) (such an Ulrich bundle is called special).

The paper under review completes the existence of special Ulrich bundles for every polarized \(K3\) surface. Let \(X\) be an integral projective surface with \(\omega_X \cong \mathcal O_X\) and \(H^1(\mathcal O_X)=0\), and let \(H = \mathcal O_X(1)\) be a very ample divisor on \(X\) which defines an embedding \(X \hookrightarrow \mathbb{P}^g\). Using Serre’s corrsepondence and an elementary modification with respect to a point, the author constructs a simple sheaf \(\mathcal E^{\eta}\) with \(c_1 (\mathcal E^{\eta}) = H\), \(c_2 (\mathcal E^{\eta}) = g+3\), \(H^{*} (\mathcal E^{\eta}) = 0\) (Lemma 2–4). Even though \(\mathcal E^{\eta}\) itself is not an Ulrich bundle, its general deformation becomes an Ulrich bundle (Lemma 5). The author also describes the moduli space of stable Ulrich bundles of any even rank on \(X\) (Theorem 7). Note that this construction is similar to construction of ACM/Ulrich bundles on Fano threefolds [M. C. Brambilla and D. Faenzi, Mich. Math. J. 60, No. 1, 113–148 (2011; Zbl 1223.14047)]

Reviewer: Yeongrak Kim (Saarbrücken)

### MSC:

14J60 | Vector bundles on surfaces and higher-dimensional varieties, and their moduli |

13C14 | Cohen-Macaulay modules |

14J28 | \(K3\) surfaces and Enriques surfaces |

13H10 | Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.) |

### Citations:

Zbl 1069.14019; Zbl 1248.14058; Zbl 1390.14130; Zbl 1274.14050; Zbl 1387.14088; Zbl 1223.14047; Zbl 1435.14042
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\textit{D. Faenzi}, Algebra Number Theory 13, No. 6, 1443--1454 (2019; Zbl 1436.14081)

### References:

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[4] | 10.1007/s40879-017-0154-4 · Zbl 1390.14130 |

[5] | 10.1307/mmj/1301586307 · Zbl 1223.14047 |

[6] | 10.1142/S0129167X17500616 · Zbl 1435.14042 |

[7] | 10.1007/978-981-10-5648-2_6 · Zbl 1376.14045 |

[8] | ; Coskun, Doc. Math., 17, 1003 (2012) |

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