Classical formulae on projective surfaces and $3$-folds with ordinary singularities, revisited

TL;DR

This paper revisits classical enumerative formulas for projective surfaces and 3-folds with singularities, utilizing universal polynomials to compute invariants like weighted Euler characteristics, and provides new computational examples.

Contribution

It introduces a modern approach using universal polynomials to re-derive and compute classical enumerative formulas for singular projective varieties.

Findings

Revised classical formulas for surfaces and 3-folds with singularities.

Explicit computations of weighted Euler characteristics.

Application of universal polynomials to enumerative geometry.

Abstract

As an application of universal polynomials for local and multi-singularities of maps, we revisit classical enumerative formulae of Salmon-Cayley-Zeuthen for projective surfaces and analogous formulae of Segre-(B.)Severi-Roth for projective $3$-folds. In particular, several examples of actual computation are given using universal polynomials for computing weighted Euler characteristics of singularity loci.