Fulton–Hansen connectedness theorem

In

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 ${\displaystyle f\colon Z\to P^{n}\times P^{n}}$ 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 ${\displaystyle Z=V\times W}$, with ${\displaystyle f}$ the natural inclusion.

• Zariski's connectedness theorem
• Grothendieck's connectedness theorem
• Deligne's connectedness theorem

• JSTOR 1971249
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• ISBN 3-540-22534-X
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• PDF lectures with the result as Theorem 15.3 (attributed to Faltings, also)