EXAMPLES OF SMOOTH SURFACES IN ℙ³ WHICH ARE ULRICH-WILD

Abstract

Abstract. Let F ⊆ P 3 be a smooth surface of degree 3 ≤ d ≤ 9 whose equation can be expressed as either the determinant of a d × d matrix of linear forms, or the pfaffian of a (2d) × (2d) matrix of linear forms. In this paper we show that F supports families of dimension p of pairwise non-isomorphic, indecomposable, Ulrich bundles for arbitrary large p.

Stability of rank two Ulrich bundles on projective $K3$ surfaces

“…Another almost immediate application of this existence result is that such a surface also supports families of arbitrary dimension of non-isomorphic, indecomposable Ulrich bundles (see the note [7]). In [8] other interesting families of initialized aCM bundles of rank 2 are constructed: their existence implies that F is quadratic pfaffian, i.e.…”

The Representation Type of Determinantal Varieties

“…Also in the case (n − c, t, c) = (2, 2, 3) Theorem 5.4 does not apply since dim Ext 1 O X (L 2 , L 1 ) = 2; it corresponds to a quartic scroll in P 5 which has tame representation type (see [20]). Note that the case n = 3 of (4) is also proved in [10].…”

Geometry of certain Brill-Noether locus on a very general sextic surface and Ulrich bundles

“…In that very survey article it's mentioned that there are no specific results for Ulrich bundles on surfaces in P 3 of degree d ≥ 5. In [8] it's shown that for smooth surfaces in P 3 with degree 3 ≤ d ≤ 9, the surface supports families of dimension p of pairwise non-isomorphic, indecomposable, Ulrich bundles for arbitrarily large p. Also in [9] and [35] rank 2 ACM bundles on a general quintic and sextic surface are classified. This motivates us to study weakly Ulrich bundles and Ulrich bundles on a very general sextic hypersurface in P 3 and its implication to the non-emptiness question of the Brill-Noether theory on such surfaces.…”

“…More precisly, we will collect facts regarding coherent systems from [25], we will recall necessary useful facts regarding algebraic curves from [2] and other sources, necessary results regarding sheaves on surface and Cayley-Bacharach property from [26] , results regarding the geometry of moduli of rank 2 stable bundles on a very general sextic surface in P 3 from [34]. Finally, we will recall facts regarding Ulrich bundles on surfaces from [13], [7], [14], [8] and other sources. In section 3, for convenience we briefly outline the construction of the Brill-Noether locus W r k,H following [15], [17].…”