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Hodge numbers of nodal double octics. (English) Zbl 0958.14032

Let \(\pi:X\to \mathbb{R}^3\) be the double covering branched over a hypersurface \(B\) of even degree \(d\) with ordinary double nodes as the only singularities. If \(\widehat X\) is a smooth threefold (double solid) obtained from \(X\) by a small resolution of its nodes, then C. H. Clemens [Adv. Math. 47, 107-230 (1983; Zbl 0509.14045)] found the formula for the Hodge numbers of such threefolds, depending non only on the number of nodes but also on a non-negative integer describing the configuration of nodes called the defect. The author gives a geometric interpretation of the defect. He proves that it is equal to the number of independent divisors on \(\widehat X\), which do not come from a divisor on \(\mathbb{P}^3\). In fact he shows that the defect depends on the geometry of contact surfaces \(S\) (to \(B)\), which are determined by the pushed-down irreducible non-symmetric divisors in \(\widehat X\).

MSC:

14J30 \(3\)-folds
14B05 Singularities in algebraic geometry
14H20 Singularities of curves, local rings

Citations:

Zbl 0509.14045
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References:

[1] DOI: 10.1016/0001-8708(83)90025-7 · Zbl 0509.14045 · doi:10.1016/0001-8708(83)90025-7
[2] Endrass S., J. of Alg. Geom 6 pp 325– (1997)
[3] Griffiths P., Principles of Algebraic Geometry (1978) · Zbl 0408.14001
[4] Hartshorne R., Algebraic Geometry (1977)
[5] Jaffe D.B., J. Alg.Geom 6 pp 151– (1997)
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