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:
“I do not want to be covetous, but I think I speak the minds of many a wife and mother when I say I would willingly work as hard as possible all day and all night, if I might be sure of a small profit, but have worked hard for twenty-five years and have never known what it was to receive a financial compensation and to have what was really my own.”
—Emma Watrous, U.S. inventor. As quoted in Feminine Ingenuity, ch. 8, by Anne L. MacDonald (1992)
“Good resolutions are useless attempts to interfere with scientific laws. Their origin is pure vanity. Their result is absolutely nil. They give us, now and then, some of those luxurious sterile emotions that have a certain charm for the weak.... They are simply cheques that men draw on a bank where they have no account.”
—Oscar Wilde (1854–1900)