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Nodal quintics in \(\mathbb P^4\). (English) Zbl 0697.14027
Arithmetic of complex manifolds, Proc. Conf., Erlangen/FRG 1988, Lect. Notes Math. 1399, 48-59 (1989).
[For the entire collection see Zbl 0675.00006.]
Generalizing a construction of Hirzebruch, the authors construct quintic threefolds in \({\mathbb{P}}^ 4\) with many nodes \((=ordinary\) double points). The constructed examples are of the form \(F(x,y)-G(u,v)=0\quad (affine\quad equation)\) where F and G are of degree \( 5.\) E.g. if \(G(x,y)=(x+1)(y^ 2-(x-2)^ 2/3)(x^ 2+y^ 2-8/5)\) \((G=0\) defines a union of a circle and a triangle) then \(G(x,y)-G(u,v)=0\) has 118 nodes; if \(F_ c(x,y)=(x+c)(y^ 4-y^ 2(2x^ 2-2x+1)+(x^ 2+x-1)^ 2/5)\) \((F_ c=0\) defines a skew pentagon which is regular if \(c=)\) then \(F_{-2}(x,y)-F_{-2}(u,v)=0\) has 120 nodes. Hirzebruch used the regular pentagon \(F_{}\) and obtained 126 nodes which is still the record. It appears as a special member of a 1-parameter family of quintics where the general member has 120 nodes.
It is an interesting question whether there are small resolutions of such a quintic which are not projective algebraic. To decide this it is essential to know the divisors on V which pass through the nodes and which are not homologous to a multiple of the generic hyperplane section. The number of independent such divisors is called the defect. Although the authors cannot decide the projectivity of some small resolutions of their examples, they compute the defect by an interesting method based on computing the absolute value of the eigenvalues of Frobenius acting on étale cohomology and the Lefschetz fixed-point formula over a suitable finite field.
Reviewer: G.-M.Greuel

14J30 \(3\)-folds
14B05 Singularities in algebraic geometry
14E15 Global theory and resolution of singularities (algebro-geometric aspects)
14N05 Projective techniques in algebraic geometry