A geometric construction of elliptic conic bundles in $P^4$

Abstract

An elliptic conic bundle in P4 is a smooth surface S ⊂ P4 with a map S → C onto an elliptic curve C, whose general fibers are embedded as smooth conic sections in P4. In this note I give a construction of the general elliptic conic bundle in P4 via Cremona transformations. More precisely I construct the general elliptic conic bundle which has a minimal model isomorphic to a smooth quintic elliptic scroll in P4, i.e. a scroll with minimal selfintersection of a section equal to 1. I do not prove that every conic bundle have a minimal model like that, but that the general one does. For any smooth elliptic quintic scroll S5 in P4 we find reducible curves G∪L consisting of a rational normal quartic curve G which meet the scroll in ten points and a line L which is secant to G and a ruling in the scroll. The quadric hypersurfaces passing through G∪L define a Cremona transformation of P4, and the image under this transformation of the elliptic scroll is a smooth conic bundle in P4. Conversely, any elliptic conic bundle in P4 which have a minimal model isomorphic to a scroll S5, contains reducible curves C2 ∪ C3 consisting of a curve C2 of degree 6 and genus 4 and a curve C3 of degree 9 and genus 6 meeting C2 in six points three on each of a pair of skew lines. The curve C2 lies on a quadric surface, while the curve C3 lies on a rational cubic scroll which meet the quadric surface along the two lines. The cubic hypersurfaces through the union of the quadric and the cubic surface defines a Cremona transformation inverse to the one above, and the image of the elliptic conic bundle is a smooth elliptic scroll of degree 5. A conic bundle S is not minimal, and the points blown up on a minimal model S5 to obtain a surface S may not be chosen in general position. The Cremona transformation gives an extrinsic description in the P4 of S5 of how the points chosen as part of the intersection of S5 and a rational normal quartic curve gives rise to a surface S. An intrinsic characterization of the eight points blown up is given in section 3 and compared with the Cremona construction.