Ulrich bundles on special cubic fourfolds of small discriminants (Q6612143)

From the introduction: ``Let \(X \subset \mathbb{P}^N\) be a projective variety of dimension \(n\), and let \(E\) be a coherent sheaf on \(X\). \(E\) is called arithmetically Cohen-Macaulay (ACM for short) if \(E\) is locally Cohen-Macaulay, and \(E\) has no intermediate cohomology, that is, \(H^i(X, E(j)) = 0\)\Nfor every \(0 < i < n\) and \(j \in \mathbb{Z}.\) \(E\) is called Ulrich if \(E\) has the same cohomological behavior as the structure sheaf\Nof \(\mathbb{P}^n\), that is, \(H^i(X, E(-j)) = 0\) for every \(i \in \mathbb{Z}\) and\N\(1\leq j \leq n.\) Thus, a coherent sheaf \(E\) on a projective variety\N\(X \subset \mathbb{P}^N \) of dimension n is called Ulrich if\Nit satisfies \(\pi_*E \cong \mathcal{O}^{\oplus t}_{\mathbb{P}^n}\) for some \(t \in \mathbb{Z}\), where \(\pi: X \to \mathbb{P}^n\) is a finite linear projection. Consequently, such a sheaf has the same cohomology table as a trivial vector bundle over \(\mathbb{P}^n\) (of rank \(t\)). In\Nparticular, the existence of an Ulrich sheaf implies that the cone of cohomology tables of\Ncoherent sheaves on \(X\) coincides with the cone of cohomology tables of coherent sheaves\Non \(\mathbb{P}^n\). There are several important questions about the structure of this cone, which may\Nprovide decompositions of the cohomology table of a given coherent sheaf on a polarized pair \((X, \mathcal{O}_X(1))\). Eisenbud and\NSchreyer asked whether an Ulrich sheaf exists for any \(X,\) whose positive answer is often\Ncalled a conjecture of \textit{D. Eisenbud} and \textit{F.-O. Schreyer} [Abel Symp. 6, 35--48 (2011; Zbl 1248.14058)] and \textit{D. Faenzi} [Algebra Number Theory 13, No. 6, 1443--1454 (2019; Zbl 1436.14081)]. When \(X\) is nonsingular, an Ulrich sheaf must be locally free, so we call it an Ulrich bundle.''\N\N``We focus on the case of smooth cubic fourfolds. Unlikely cubic hypersurfaces of dimension \(\leq 3,\) not all smooth cubic fourfolds have the same Ulrich complexity. For instance,\Nif a smooth cubic fourfold \(X\) contains a del Pezzo surface of degree \(5\) then \(X\) has an\NUlrich bundle of rank \(2,\) however, not all smooth cubic fourfolds contain such a surface. In the very general situation (so that\N\(X\) is a smooth cubic fourfold and does not contain such special surfaces), it is straight-forward that the rank \(r\) of any Ulrich bundle on \(X\) must be divisible by \(3\) and \(r \geq 6\) [\textit{Y. Kim} and \textit{F.-O. Schreyer}, J. Pure Appl. Algebra 224, No. 8, Article ID 106346, 13 p. (2020; Zbl 1437.13023), Proposition \(2.5\)]. Note also that any smooth cubic fourfold has Ulrich bundles of\Nrank \(6\) [\textit{D. Faenzi} and \textit{Y. Kim}, Comment. Math. Helv. 97, No. 4, 691--728 (2022; Zbl 1517.14029), Theorem \(1\)] and a general cubic fourfold has Ulrich bundles of rank \(9\) [\textit{L. Manivel}, Commun. Algebra 47, No. 2, 706--718 (2019; Zbl 1436.14082), Proposition \(2.2\)]; \textit{Y. Kim} and \textit{F.-O. Schreyer}, J. Pure Appl. Algebra 224, No. 8, Article ID 106346, 13 p. (2020; Zbl 1437.13023), Theorem \(1.1\)], hence, a general cubic fourfold has Ulrich bundles\Nof rank \(3k\) for every \(k \geq 2.\)\NIt is thus interesting to observe smooth cubic fourfolds having Ulrich bundles of rank\N\(r\) not obtained by those rank \(6\) or rank \(9\) Ulrich bundles. The most well-understood\Nexamples are Pfaffian cubic fourfolds which can be also characterized as cubic fourfolds\Ncontaining a del Pezzo surface \(Y_5\) of degree \(5\) [\textit{A. Beauville}, Mich. Math. J. 48, 39--64 (2000; Zbl 1076.14534), Proposition \(9.2\)]. Very\Nrecently, Truong and Yen reported a few cases of special cubic fourfolds having Ulrich\Nbundles of rank \(3\) or \(4\) [\textit{T. Le Hoang} and \textit{Y. N. Hoang}, Manuscr. Math. 174, No. 1--2, 243--267 (2024; Zbl 1539.13031), Theorem \(4.6, 4.11\)] by finding a linear matrix factorization\Nvia computer-assisted proofs. We provide computer-based construction of\NUlrich bundles of small ranks over some special cubic fourfolds\Nof small discriminants via deformation theory. First, we\Nconstruct a simple sheaf whose Chern classes are the same\Nas an Ulrich bundle of the same rank (if exists) as the syzygy\Nsheaf of an Ulrich sheaf on a surface contained in a special\Ncubic fourfold. We observe that a general deformation of this\Nsimple sheaf becomes Ulrich in several cases. We also provide\Nsome experimental constructions of Ulrich bundles on special\Ncubic fourfolds of small discriminants.''