Irregular Elliptic Surfaces of Degree 12 in the Projective Fourspace

The purpose of this paper is to give two different constructions of irregular surfaces of degree 12 in P4. These surfaces are new and of interest in the classification of smooth non-general type surfaces in P4. This classification problem is motivated by the theorem of Ellingsrud and Peskine [9] which says that the degree of smooth non-general type surfaces in P4 is bounded. Moreover the new family is one of a few known families of irregular surfaces in P4. The other known irregular smooth surfaces in P4 are the elliptic quintic scrolls, the elliptic conic bundles [2], the bielliptic surfaces of degree 10 [16] and degree 15 [4], the minimal abelian surfaces of degree 10 [11] and the non-minimal abelian surfaces of degree 15 (cf. [3]) up to pull-backs of these families by suitable finite maps from P4 to P4 itself. The first author came across the irregular elliptic surfaces when studying a stable rank three vector bundle E on P4 with Chern classes (c1, c2, c3) = (5, 12, 12) (see [1] for the explicit construction of this bundle). The dependency locus of two sections of E is a smooth surface of the desired type. Our first construction uses monads. The basic idea of monads is to represent a given coherent sheaf as a cohomology sheaf of a complex of simpler vector bundles. A useful way to construct monads is Horrocks’ technique of killing cohomology. As we will see in § 2.2, for the ideal sheaf of a smooth surface X in P4 this technique is closely related to the graded finite length modules H∗IX = ⊕ m∈Z H i(P4, IX(m)), i = 1, 2, over the polynomial ring C[x0, . . . , x4] called the Hartshorne-Rao modules of X. To construct an irregular elliptic surface, we construct the ideal sheaf via its Hartshorne-Rao modules. The second construction uses liaison, and reduces the construction of an irregular surface as above to the construction of a simpler locally complete intersection surface. More precisely we show that the elliptic irregular surface of the first construction is linked (5, 5) to a reducible surface of degree 13. This reducible surface consists of a singular rational cubic scroll surface in a hyperplane of P4 and a smooth general type surface of degree 10 and sectional genus 10 such that they intersect in three disjoint (singular) conics on the cubic surface. We then show that these reducible surfaces can be constructed directly, in a slightly more general form. In fact we construct a cubic Del Pezzo surface X0 and a smooth general type surface T of degree 10 and sectional genus 10 such that T ∩ X0 is the disjoint union of three smooth conics and show that the general surface linked (5, 5) to T ∪X0 is a smooth irregular elliptic surface. The family of irregular elliptic surfaces constructed via liaison includes the family of irregular elliptic surfaces constructed via monad as a special case.