The Unicity of a Blow-down.
Master thesis
Permanent lenke
https://hdl.handle.net/11250/3089805Utgivelsesdato
2023Metadata
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Sammendrag
In algebraic geometry, projective varieties are classified up to isomorphism. Thisinvolves classifying the varieties up to birational equivalence and classifyingnonsingular varieties in each equivalence class up to isomorphism. Singularprojective varieties are modified to less singular or nonsingular ones by blowingup the singularities. A blow up map contracts or blows down an exceptionaldivisor to a curve or a point. In surfaces, such maps (blow-downs) exist and are unique up toisomorphism and by the Castelnuovo contraction criterion, any curve that can beblown down is a -1 curve. In higher dimensional varieties, contractionmorphisms/blow-downs are uniquely determined by the extremal rays which they contract. Inthis thesis, we present a result due to Lascu [1] on the uniqueness of ablow-down. Precisely, Lascu shows that any birational morphism f : X −→Sthat contracts a divisor D ⊂ X to a subvariety Y ⊂ S is a blow-down if and only ifS is a nonsingular variety, D is a closed nonsingular divisor.