Small Resolutions
A resolution of singularities
of a complex variety Y is called a small resolution if for every r>0, the space of points of Y where the fiber has dimension r is of codimension greater than 2r. Roughly speaking, this means that most fibers are small. In this case the morphism induces an isomorphism from the (intersection) homology of X to the intersection homology of Y (with the middle perversity).
There is a variety with two different small resolutions that have different ring structures on their cohomology, showing that there is in general no natural ring structure on intersection (co)homology.
Read more about this topic: Intersection Homology
Famous quotes containing the words small and/or resolutions:
“... it is an uneasy lot at best, to be what we call highly taught and yet not to enjoy: to be present at this great spectacle of life and never to be liberated from a small hungry shivering self—never to be fully possessed by the glory we behold, never to have our consciousness rapturously transformed into the vividness of a thought, the ardour of a passion, the energy of an action, but always to be scholarly and uninspired, ambitious and timid, scrupulous and dim-sighted.”
—George Eliot [Mary Ann (or Marian)
“No woman ever yet either reasoned or acted long together consequentially; but some little thing, some love, some resentment, some present momentary interest, some supposed slight, or some humour, always breaks in upon, and oversets their most prudent resolutions and schemes.”
—Philip Dormer Stanhope, 4th Earl Chesterfield (1694–1773)