Do these definitions of irreducibility of algebraic sets coincide?

Question

I am reading the book Numerically solving polynomials systems with Bertini in which they define a manifold point $p^* = (p_1^*,\ldots,p_m^*)$ of an algebraic set $X$ to be a point in $X$ with an open neighborhood $U\subset X$ such that for some mapping $\Phi(z_1,\ldots,z_m)$, $\Phi$ restricted to $U$ maps $U$ bijectively onto a neighborhood of the origin in $\mathbb{C}^k$ for some $k$. The set of manifold points of $X$ is denoted $X_{\text{reg}}$.

Now they say an affine complex algebraic set $X$ is irreducible if $X_{\text{reg}}$ is connected, i.e. $X_{\text{reg}}$ cannot be written as the union of two disjoint non-empty open subsets in $X_{\text{reg}}$.

However, in my algebraic geometry class, which is based on the book of Hartsthorne. An algebraic set $X$ is irreducible if it cannot be expressed as the union of two proper non-empty closed subsets of $X$.

I have not seen any mention of manifold points in Hartsthorne so far and am having trouble understanding which points of an algebraic set are manifold points (or rather which points are not) and therefore how these two definitions of irreducibility coincide.

Answer 1

The result is that these definitions are equivalent for algebraic varieties over $\Bbb C$. (I use "variety" in the general sense - I do not require a variety to be irreducible.)

"Manifold points" of $X$ are better known as regular points (as hinted at with the notation $X_{reg}$). This means that the local rings $\mathcal{O}_{X,x}$ of these points $x$ are regular local rings. In particular, a regular local ring has only one minimal prime, so a regular point $x$ lies on exactly one irreducible component via the correspondence between irreducible components passing through $x$ and minimal primes of $\mathcal{O}_{X,x}$. As a consequence, every point which is in the intersection of two irreducible components cannot be regular.

This gives us our equivalence between the two characterizations of irreducible: if we have a variety with multiple irreducible components, then the manifold points of these distinct irreducible components are disjoint and thus the manifold points of the whole variety are not connected. Conversely, if $X$ is irreducible, then the set of non-manifold points is algebraic codimension one (or real codimension two) and therefore it's removal cannot cause $X$ to become disconnected.

Hartshorne chapter I section 5 deals with nonsingular varieties and would be a good reference for you to check out - it contains a couple characterizations of when a point is regular that you can actually use (the Jacobian criterion is the big one, and it's one of the first results of the section).