Yeongrak Kim

We investigate a geometric criterion for a smooth curve [ ] of genus [ ] and degree [ ] to be described as the zero locus of sections in an Ulrich bundle of rank [ ] on a del Pezzo threefold [ ]. The main challenge is to read off the Pfaffian quadrics defining [ ] from geometric structures of [ ]. We find that this problem is related to the existence of a special rank-2 vector bundle on [ ] with trivial resonance. It gives a description of the image of the rational map that appeared in a work of Ciliberto–Flamini–Knutsen, for the case of degree [ ] del Pezzo threefolds. From an explicit calculation of the Betti table of such a curve, we also deduce the uniqueness of the del Pezzo threefold containing a given curve.

Using a part of XJC-correspondence by Pirio and Russo, we classify cubic forms f f whose Hessian matrices induce matrix factorizations of themselves. When it defines a reduced hypersurface, it satisfies the “secant–singularity” correspondence, that is, it coincides with the secant locus of its singular locus. In particular, when f f is irreducible, its singular locus is one of the four Severi varieties.

We discuss the Borisov–Nuer conjecture in connection with the canonical maps from the moduli spaces of polarized Enriques surfaces with fixed to the moduli space of polarized K3 surfaces of genus g with , and we exhibit a naturally defined locus . One direct consequence of the Borisov–Nuer conjecture is that would be contained in a particular Noether–Lefschetz divisor in , which we call the Borisov–Nuer divisor and we denote by . In this short note, we prove that is non‐empty whenever is divisible by 4. To this end, we construct polarized Enriques surfaces , with divisible by 4, which verify the conjecture. In particular, when we consider the moduli space of (numerically) polarized Enriques surfaces which contains such , the conjecture also holds for any other polarized Enriques surface contained in the same moduli.

In this paper, we compute an explicit matrix factorization of a rank 9 Ulrich sheaf on a general cubic hypersurface of dimension at most 7, whose existence was proved by Manivel. Instead of using invariant theory, we use Shamash's construction with a cone over the spinor variety. We also describe an algebro-geometric interpretation of our matrix factorization which connects the spinor tenfold and the Cartan cubic.

We construct an Ulrich bundle on the blowup at a point when the original variety is embedded by a sufficiently positive linear system and carries an Ulrich bundle. In particular, we describe the relation between special Ulrich bundles on the blown-up surfaces and the original surfaces.