Smooth fundamental divisors from canonical double cuts and first jets

Let Z be the reduced base locus of V. By [F2], a general member is smooth off Z. A section vanishes on Z, so its differential there is independent of the local A-trivialization up to the corresponding unit: the derivative of the transition unit is multiplied by the vanishing section. Thus there is an intrinsic map V tensor $O_Z\to Omeg{a}_X\mathrm{\mid }Z$ tensor A|Z.

Stratify Z into finitely many smooth irreducible locally closed strata on which this derivative map has constant rank and locally free image. To construct the strata, take its determinantal rank loci; on the dense smooth open of each irreducible component of each difference the image and kernel are bundles. Remove those opens and repeat on their smaller-dimensional closed complements. Finite Noetherian induction terminates. Fix one resulting stratum of dimension d and derivative image bundle F of rank r.

For $r>0$, the sections having zero derivative at a point of this stratum form the projective kernel bundle I. Its dimension is d+dim P(V)-r, unless its fibres are empty. If its image in P(V) is not dense, a general section has no singular point on this stratum. If the image is dense, dimension gives $d\ge r$.

In the latter case choose a general s1 that has a zero of its derivative at a point z of the stratum. Choose s2 general with ds2(z) nonzero and with the actual double cut T canonical as hypothesized. Such a choice is possible: derivative evaluation at z has positive rank, and the canonical-doublecut premise holds for general s2 after fixing a general s1. Shrink the stratum near z so ds2 is nowhere zero in F.

The locus where ds1 and ds2 are dependent is the zero locus of the image of ds1 in F/O ds2, of rank r-1. It contains z and is locally defined by at most r-1 functions on the smooth d-dimensional stratum. Its local dimension at z is at least $d-r+1\ge 1$ by [F3].

Every point of that dependence locus lies in Z and hence in T; the two defining gradients have rank at most one. Because T is the actual codimension-two complete intersection in smooth X, [F2] makes all those points singular on T. This gives a positive-dimensional singular locus of the normal surface T, contradicting [F2]. Thus the dense-incidence alternative cannot occur on any positive-rank stratum.

On a rank-zero stratum every section and every first derivative vanish, so every double cut is singular there. The actual normal-doublecut premise excludes every positive-dimensional such stratum. There remain only finitely many fixed rank-zero points.

At each remaining point p, a local trivialization of A gives $s1=f1$ and $s2=f2$ with f1,f2 in $m_p^2$ for every pair of sections. A general pair is a normal canonical codimension-two surface by hypothesis. This is precisely forbidden by [F1]. Hence no fixed rank-zero point exists.

There are finitely many strata. For each positive-rank stratum the incidence image has proper closure, and the rank-zero strata have just been excluded. Choose a general section outside their finitely many proper closures and the off-base Bertini exceptional locus. Its divisor is smooth everywhere on X, proving the claim.