Intersections via resolutions

We investigate the viability of defining an intersection product on algebraic cycles on a singular algebraic variety by pushing forward intersection products formed on a resolution of singularities. For varieties with resolutions having a certain structure (including all varieties over a field of characteristic zero), we obtain a stratification which reflects the geometry of the centers and the exceptional divisors. This stratification is sufficiently fine that divisors can be intersected with $r$-cycles (for $r \geq 1$), and 2-cycles can be intersected on a fourfold, provided their incidences with the strata are controlled. Similar pairings are defined on a variety with one-dimensional singular locus.