6.6: Hypervarieties

Pure codimension 1 subvarieties are particularly nice. Sometimes pure codimension 1 subvarieties are called hypervarieties. Back in Section 1.6 we proved the following theorem (Theorem 1.6.2). We restate it in the language of varieties.

Theorem \(\PageIndex{1}\)

Let \(U \subset \mathbb{C}^n\) be a domain and \(f \in \mathcal{O}(U)\). Then \(Z_f\) is empty, a subvariety of pure codimension 1, or \(Z_f = U\). Furthermore, if \(Z_f\) is a pure codimension 1 subvariety, then \((Z_f)_{\mathit{reg}}\) is open and dense in \(Z_f\).

We can improve on that slightly. Let us prove that \((Z_f)_{\mathit{sing}}\) is (locally) contained in the zero set of some holomorphic function that is not zero on any nonempty open subset of \(Z_f\). It suffices to assume \(0 \in Z_f\) and to prove this locally near the origin. As usual, after possibly a linear change of coordinates, we assume that near the origin \(Z_f\) is contained in some \(U' \times D \subset \mathbb{C}^{n-1} \times \mathbb{C}\) for a disc \(D\) and \(Z_f\) does not intersect \(U' \times \partial D\). Apply Theorem 6.3.1 to find the discriminant function \(\Delta\) and the discriminant \(E = \Delta^{-1}(0)\). Above a neighborhood of each point in \(U' \setminus E\), the set \(Z_f\) is union of \(m\) distinct graphs of holomorphic functions (see Proposition 6.3.2), so those points are regular. Above each point of \(E\) there are only finitely many points of \(Z_f\), and \(E\) is nowhere dense in \(U'\). So the function \((z',z_n) \mapsto \Delta(z')\) does not vanish on any nonempty open subset of \(Z_f\).

Theorem \(\PageIndex{2}\)

If \((X,p)\) is a germ of a pure codimension 1 subvariety, then there is a germ of a holomorphic function \(f\) at \(p\) such that \((Z_f,p) = (X,p)\). Further, \(I_p(X)\) is generated by \((f,p)\).

Proof

We already proved all the relevant pieces of the first part of this theorem.

For the second part there has to exist a germ of a function that vanishes on \((X,p)\). Assume \(p=0\), and after a linear change of coordinates assume we can apply the Weierstrass preparation theorem to the function. Taking representatives of the germs, we assume \(X\) is a pure codimension 1 subvariety of a small enough neighborhood \(U' \times D \subset \mathbb{C}^{n-1} \times \mathbb{C}\) of the origin, where \(D\) is a disc, and the function that vanishes on \(X\) is a Weierstrass polynomial \(P(z',z_n)\) defined for \(z' \in U'\), and all zeros of \(z_n \mapsto P(z',z_n)\) are in \(D\) for \(z' \in U\).

Theorem 6.3.1 applies. Let \(E \subset U'\) be the discriminant set, a zero set of a holomorphic function. On \(U' \setminus E\), there are a certain number of geometrically distinct zeros of \(z_n \mapsto P(z',z_n)\).

Let \(X'\) be a topological component of \(X \setminus ( E \times D )\). Above each point \(z' \in U' \setminus E\), let \(\alpha_1(z'),\ldots,\alpha_k(z')\) denote the distinct zeros that are in \(X'\), that is \(\bigl(z',\alpha_j(z')\bigr) \in X'\). If \(\alpha_j\) is a holomorphic function in some small neighborhood and \(\bigl(z',\alpha_j(z')\bigr) \in X'\) at one point, then \(\bigl(z',\alpha_j(z')\bigr) \in X'\) for all nearby points too. Furthermore, this means that the set \(X'\) contains only regular points of \(X\) of dimension \(n-1\).

The number of such geometrically distinct zeros above each point in \(U' \setminus E\) is locally constant, and as \(U' \setminus E\) is connected there exists a unique \(k\). Take \[F(z',z_n) = \prod_{j=1}^k \bigl( z_n-\alpha_j(z')\bigr) = z_n^k + \sum_{j=0}^{k-1} g_j(z') z_n^j .\] The coefficients \(g_j\) are well-defined for \(z \in U' \setminus E\) as they are independent of how \(\alpha_1,\ldots,\alpha_k\) are ordered. The \(g_j\) are holomorphic for \(z \in U' \setminus E\) as locally we can choose the order so that each \(\alpha_j\) is holomorphic. The coefficients \(g_j\) are bounded on \(U'\) and therefore extend to holomorphic functions of \(U'\). Hence, the polynomial \(F\) is a polynomial in \(\mathcal{O}(U')[z_n]\). The zeros of \(F\) above \(z' \in U' \setminus E\) are simple and give precisely \(X'\). By using the argument principle again, we find that all zeros above points of \(E\) are limits of zeros above points in \(U' \setminus E\). Consequently, the zero set of \(F\) is the closure of \(X'\) in \(U' \times D\) by continuity. It is left to the reader to check that all the functions \(g_j\) vanish at the origin and \(F\) is a Weierstrass polynomial, a fact that will be useful in the exercises below.

If the polynomial \(P(z',z_n)\) is of degree \(m\), then \(z' \mapsto P(z',z_n)\) has at most \(m\) zeros. Together with the fact that \(U' \setminus E\) is connected, this means that \(X \setminus (E \times D)\) has at most finitely many components (at most \(m\)). So we can find an \(F\) for every topological component of \(X \setminus ( E \times D )\). Then we multiply those functions together to get \(f\).

The fact that this \(f\) will generate \(I_p(X)\) is left as an exercise below.

In other words, local properties of a codimension 1 subvariety can be studied by studying the zero set of a single Weierstrass polynomial.

Example \(\PageIndex{1}\)

It is not true that if a dimension of a subvariety in \(\mathbb{C}^n\) is \(n-k\) (codimension \(k\)), there are \(k\) holomorphic functions that “cut it out.” That only works for \(k=1\). The set defined by \[\text{rank} \begin{bmatrix} z_1 & z_2 & z_3 \\ z_4 & z_5 & z_6 \end{bmatrix} < 2\] is a pure 4-dimensional subvariety of \(\mathbb{C}^6\), so of codimension 2, and the defining equations are \(z_1z_5-z_2z_4 = 0\), \(z_1z_6-z_3z_4 = 0\), and \(z_2z_6-z_3z_5 = 0\). Let us state without proof that the unique singular point is the origin and there exist no 2 holomorphic functions near the origin that define this subvariety. In more technical language, the subvariety is not a complete intersection.

Example \(\PageIndex{2}\)

If \(X\) is a hypervariety and \(E\) the corresponding discriminant set, it is tempting to say that the singular set of \(X\) is the set \(X \cap (E \times \mathbb{C})\), which is a codimension 2 subvariety. It is true that \(X \cap (E \times \mathbb{C})\) will contain the singular set, but in general the singular set is smaller. A simple example of this behavior is the set defined by \(z_2^2 - z_1 = 0\). The defining function is a Weierstrass polynomial in \(z_2\) and the discriminant set is given by \(z_1 = 0\). However, the subvariety has no singular points. A less trivial example is given in an exercise below.

Interestingly we also proved the following theorem. Same theorem is true for higher codimension, but it is harder to prove.

Corollary \(\PageIndex{1}\)

Let \((X,p)\) is a germ of a subvariety of pure codimension 1. Then there exists a neighborhood \(U\) of \(p\), a representative \(X \subset U\) of \((X,p)\) and subvarieties \(X_1,\ldots,X_k \subset U\) of pure codimension 1 such that \((X_j)_{\mathit{reg}}\) is connected for every \(j\), and \(X = X_1 \cup \cdots \cup X_k\).

Proof

A particular \(X_j\) is defined by considering a topological component of \(X \setminus (E \times D)\) as in the proof of Theorem \(\PageIndex{2}\), getting the \(F\), and setting \(X_j = Z_F\). The topological component is of course a connected set and it is dense in \((X_j)_{\mathit{reg}}\), which proves the corollary.

Exercise \(\PageIndex{1}\)

1. Prove that the hypervariety in \(\mathbb{C}^n\), \(n \geq 2\), given by \(z_1^2 + z_2^2 + \cdots + z_n^2 = 0\) has an isolated singularity at the origin (that is, the origin is the only singular point).
2. For any \(0 \leq k \leq n-2\), find a hypervariety \(X\) of \(\mathbb{C}^n\) whose set of singular points is a subvariety of dimension \(k\).

Exercise \(\PageIndex{2}\)

Suppose \(p(z',z_n)\) is a Weierstrass polynomial of degree \(k\) such that for an open dense set of \(z'\) near the origin \(z_n \mapsto p(z',z_n)\) has geometrically \(k\) zeros, and such that the regular points of \(Z_p\) are connected. Show that \(p\) is irreducible in the sense that if \(p = rs\) for two Weierstrass polynomials \(r\) and \(s\), then either \(r=1\) or \(s=1\).

Exercise \(\PageIndex{3}\)

Suppose \(f\) is a function holomorphic in a neighborhood of the origin with \(z_n \mapsto f(0,z_n)\) being of finite order. Show that \[f = u p_1^{d_1} p_2^{d_2} \cdots p_\ell^{d_\ell} ,\] where \(p_j\) are Weierstrass polynomials of degree \(k_j\) that have generically (that is, on an open dense set) \(k_j\) distinct zeros (no multiple zeros), the regular points of \(Z_{p_j}\) are connected, and \(u\) is a nonzero holomorphic function in a neighborhood of the origin. See also the next section, these polynomials will be the irreducible factors in the factorization of \(f\).

Exercise \(\PageIndex{4}\)

Suppose \((X,p)\) is a germ of a pure codimension 1 subvariety. Show that the ideal \(I_p(X)\) is a principal ideal (has a single generator).

Exercise \(\PageIndex{5}\)

Suppose \(I \subset \mathcal{O}_p\) is an ideal such that \(V(I)\) is a germ of a pure codimension 1 subvariety. Show that the ideal \(I\) is principal.

Exercise \(\PageIndex{6}\)

Let \(I \subset \mathcal{O}_p\) be a principal ideal. Prove the Nullstellensatz for hypervarieties: \(I_p\bigl(V(I)\bigr) = \sqrt{I}\). That is, show that if \((f,p) \in I_p\bigl(V(I)\bigr)\), then \((f^k,p) \in I\) for some integer \(k\).

Exercise \(\PageIndex{7}\)

Suppose \(X \subset U\) is a subvariety of pure codimension 1 for an open set \(U \subset \mathbb{C}^n\). Let \(X'\) be a topological component of \(X_{\textit{reg}}\). Prove that the closure \(\overline{X'}\) is a subvariety of \(U\) of pure codimension 1.