| dc.description.abstract | In algebraic geometry, projective varieties are classified up to isomorphism. This
involves classifying the varieties up to birational equivalence and classifying
nonsingular varieties in each equivalence class up to isomorphism. Singular
projective varieties are modified to less singular or nonsingular ones by blowing
up the singularities. A blow up map contracts or blows down an exceptional
divisor to a curve or a point. In surfaces, such maps (blow-downs) exist and are unique up to
isomorphism and by the Castelnuovo contraction criterion, any curve that can be
blown down is a -1 curve. In higher dimensional varieties, contraction
morphisms/blow-downs are uniquely determined by the extremal rays which they contract. In
this thesis, we present a result due to Lascu [1] on the uniqueness of a
blow-down. Precisely, Lascu shows that any birational morphism f : X −→S
that contracts a divisor D ⊂ X to a subvariety Y ⊂ S is a blow-down if and only if
S is a nonsingular variety, D is a closed nonsingular divisor. | |
