An example of algebraic cycles with nontrivial Abel-Jacobi images (Q1969476)

Let \(V\) be the projective closure of the affine variety of the form \(F(x,y)- F'(u,v)=0\), where \(F=0\), \(F'=0\) define configurations of five lines in the \((x,y)\)-- (respectively the \((u,v)-)\) plane. For a generic choice of \(F\) and \(F'\), the variety \(V\) has 101 ordinary double points as singularities. Let \(\widetilde V\) be the blowing-up of \(V\) at all of the 101 double points, and \(\ell^0_1- \ell^0_2\) the difference of two rulings in the exceptional divisor over \((0,0,0,0)\). The main result of this paper is formulated as follows: Assume the absolute purity of the 3-adic étale cohomology. Then for a suitable choice of \(F\) and \(F'\), the class of \(\ell_1^0-\ell^0_2\) has a 10-primary torsion cohomology class in \(\text{H}^4 (\widetilde V,\mathbb{Z})\), but its image under the 3-adic Abel-Jacobi map is nonzero.