Surfaces of Degree 10 in the Projective Fourspace via Linear Systems and Linkage

Ellingsrud and Peskine showed in [ElP] that there are finitely many components of the Hilbert scheme of P containing smooth surfaces not of general type. The upper bound for the degree of such surfaces has recently been reduced to 105 (cf. [BF]). Quite a bit of work has been put into trying to construct such surfaces of high degree. So far the record is 15, which coincides with the conjectural upper bound. This paper concerns the classification of surfaces of degree 10 and sectional genus 9 and 10. The surfaces of degree at most 9 are described through classical work dating from the last century up to recent years: [Ba], [Ro], [Io], [Ok], [Al], [AR]. Surfaces of degree at least 11 have been considered systematically recently in [Po]. In degree 10 there are the abelian surfaces discovered by Comessatti [Co], rediscovered by Horrocks and Mumford [HM] in the seventies as zero-sections of an indecomposable rank 2 vector bundle on P. Beside surfaces of sectional genus at least 11, which can be linked to smooth surfaces of smaller degree, other surfaces of degree 10 were considered only recently. Serrano gave examples of bielliptic surfaces of degree 10 in [S], which have sectional genus 6 like the abelian ones. The Hilbert schemes of abelian and bielliptic surfaces are now well understood (cf. [BHM], [BaM], [H], [HKW], [HL], [HV], [L], [Ram], [ADHPR]). The second author determined numerical invariants and gave some examples of surfaces with sectional genus 8,9 and 10. There are two families of surfaces of genus 8, one of rational surfaces, and one of non-minimal Enriques surfaces. A construction using syzygies of both of them is given in [DES]. The linear system of surfaces of the first family is described in [Ra], cf. also [Al], while for the second it is described in [Br]. The purpose of this paper is to describe the remaining components of the Hilbert scheme, i.e., to describe the smooth surfaces in P of degree 10 and sectional genus 9 and 10. Surfaces of type B, below, were first constructed in [DES] using syzygies. Examples of the other types were constructed in [Po] and [Ra] as parts of the authors thesis. They appear in this paper with new approaches and with