The authors derive recursive Euler class formulas that enumerate curves with δ nodes and one fixed singularity for all δ+k ≤ 8, recovering prior results and producing new codimension eight numbers.
Rational cuspidal curves on del-Pezzo surfaces
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abstract
We obtain an explicit formula for the number of rational cuspidal curves of a given degree on a del-Pezzo surface that pass through an appropriate number of generic points of the surface. This enumerative problem is expressed as an Euler class computation on the moduli space of curves. A topological method is employed in computing the contribution of the degenerate locus to this Euler class.
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Counting curves in a linear system with upto eight singular points
The authors derive recursive Euler class formulas that enumerate curves with δ nodes and one fixed singularity for all δ+k ≤ 8, recovering prior results and producing new codimension eight numbers.