Fulton–Hansen connectedness theorem

Fulton–Hansen connectedness theorem In mathematics, the Fulton–Hansen connectedness theorem is a result from intersection theory in algebraic geometry, for the case of subvarieties of projective space with codimension large enough to make the intersection have components of dimension at least 1. It is named after William Fulton and Johan Hansen, who proved it in 1979.

The formal statement is that if V and W are irreducible algebraic subvarieties of a projective space P, all over an algebraically closed field, and if {displaystyle dim(V)+dim(W)>dim(P)} in terms of the dimension of an algebraic variety, then the intersection U of V and W is connected.

More generally, the theorem states that if {displaystyle Z} is a projective variety and {displaystyle fcolon Zto P^{n}times P^{n}} is any morphism such that {displaystyle dim f(Z)>n} , then {displaystyle f^{-1}Delta } is connected, where {displaystyle Delta } is the diagonal in {displaystyle P^{n}times P^{n}} . The special case of intersections is recovered by taking {displaystyle Z=Vtimes W} , with {displaystyle f} the natural inclusion.

See also Zariski's connectedness theorem Grothendieck's connectedness theorem Deligne's connectedness theorem References Fulton, William; Hansen, Johan (1979). "A connectedness theorem for projective varieties with applications to intersections and singularities of mappings". Annals of Mathematics. 110 (1): 159–166. doi:10.2307/1971249. JSTOR 1971249. Lazarsfeld, Robert (2004). Positivity in algebraic geometry, Vol. I. Berlin: Springer. ISBN 3-540-22533-1. Lazarsfeld, R. K. (2004). Positivity in algebraic geometry, Vol. II. ISBN 3-540-22534-X. External links PDF lectures with the result as Theorem 15.3 (attributed to Faltings, also) Categories: Intersection theoryTheorems in algebraic geometry