Abstract
Let S be a regular surface endowed with a very ample line bundle mathcal O_S(h_S). Taking inspiration from a very recent result by D. Faenzi on K3 surfaces, we prove that if {mathcal O}_S(h_S) satisfies a short list of technical conditions, then such a polarized surface supports special Ulrich bundles of rank 2. As applications, we deal with general embeddings of regular surfaces, pluricanonically embedded regular surfaces and some properly elliptic surfaces of low degree in mathbb {P}^{N}. Finally, we also discuss about the size of the families of Ulrich bundles on S and we inspect the existence of special Ulrich bundles on surfaces of low degree.
Highlights
Let PN be the projective space of dimension N over an algebraically closed field k of characteristic 0
We say that a sheaf E on X is Ulrich (with respect to OX (h X )) if hi X, E(−i h X ) = h j X, E(−( j + 1)h X ) = 0, for each i > 0 and j < dim(X )
It follows that E := Fpg(S)(hS) is a special Ulrich bundle of rank 2: in the paper we show that it has all the properties listed in the statement of Theorem 1.1
Summary
Let PN be the projective space of dimension N over an algebraically closed field k of characteristic 0. If X ⊆ PN is a variety, i.e. an integral closed subscheme, it is naturally endowed with the very ample line bundle OX (h X ) := OPN (1) ⊗ OX. Ulrich bundles on a variety X have many properties: we refer the interested reader to [23], where the authors raised the following questions. Is every variety (or even scheme) X ⊆ PN the support of an Ulrich sheaf? When C is a curve, i.e. a smooth variety of dimension 1, the above questions have very easy answers: if g is the genus of C and L ∈ Picg−1(C) satisfies h0 C, L = 0, L(hC ) is an Ulrich line bundle. The interested reader can refer to the recent survey [10] for further results
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