When is the intersection of two irreducible variety irreducible?

Question

I am learning AG. At first, I thought the intersection of two irreducible varieties was irreducible. This isn't true in general. For instance, take the parabola and a general line. They will intersect at two points, so it isn't irreducible. I am curious, though, when is the intersection irreducible?

Answer 1

A major theorem in this direction is Bertini's theorem, which says that if $X$ is a smooth projective variety over an algebraically closed field, then the intersection of $X$ with a general hyperplane $H$ is smooth. If $\dim X \geq 2$, then $X \cap H$ is connected, so this also implies $X \cap H$ is irreducible (since a reducible connected variety is singular where the irreducible components intersect). Here "general" means that the set of hyperplanes for which this is true form a dense open subset of the space of all hyperplanes in the ambient projective space.

Generalizations of Bertini's theorems have been proved in various settings, but a common theme is that there's always some sort of "general position" requirement, where a "sufficiently general" intersection is smooth and/or irreducible, but with the possibility of exceptional intersections that fail to be irreducible.

For example, consider the hyperbolic paraboloid defined by $z = x^2 - y^2$ in $\mathbb{A}^3$. If we intersect this with the plane $z = c$ for $c \neq 0$, then we get a smooth conic curve. But if we intersect with $z = 0$, we get a union of two lines intersecting at the origin. Similar phenomena occur in many cases, and if you want to know if a specific intersection is irreducible, you have to look at specific properties of the varieties in question.