# 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

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 is a projective variety and is any morphism such that , then is connected, where is the diagonal in . The special case of intersections is recovered by taking , with 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.*Positivity in algebraic geometry, Vol. II*. 2004. ISBN 3-540-22534-X.