# "path-connectedness" of an algebraic variety

Let $X$ be an irreducible algebraic variety over a field (supposed to be algebraically closed if necessary). How to proove that any two closed points of $X$ can be connected by a finite number of curves ("connected" means in the image of a morphism from a nonsingular curve to $X$)?

• I'm not an expert at all, but you probably need your algebraic variety to be irreducible or something like that. Otherwise look at the scheme associated to the equation $(x^2+y^2-1)(x^2+y^2-2)=0$, which I'm 95% sure is a (reducible) algebraic variety. Commented Oct 30, 2015 at 15:30
• Thanks for your remark. An algebraic variety was always irreducible in my definition. I edited the question to avoid confusion.
– user151873
Commented Oct 30, 2015 at 15:34

Assume $X$ is projective for definiteness. Here is a lovely argument by C. P. Ramanujam (see his collected works). Take the two points of interest, blow them up. The resulting variety is projective, irreducible and thus by Bertini, a general hyperplane section is irreducible. Then, its image is an irreducible variety in $X$ and since a hyperplane meets all divisors, it should meet both the exceptional divisors and thus the image should contain both the points you started with. Dimension has dropped, continue till you reach a curve. So, the upshot is that any two points can be connected by an irreducible curve.