Finite projective schemes in linearly general position.

*(English)*Zbl 0804.14002If \(X\) is a reduced irreducible variety of codimension \(c\) in \(\mathbb{P}^ r\) over an algebraically closed field \(F\) of characteristic 0, then a general plane of dimension \(c\) meets \(X\) in a set of reduced points in linearly general position; that is, no \(k+2\) of them are contained in a \(k\)-plane for \(k<c\). For this reason, reduced sets of points in linearly general position play a significant role in many arguments of algebraic geometry, perhaps most notably those of Castelnuovo theory, which gives a bound on the genus of a variety in terms of its degree. In certain applications, however, it is desirable to extend the theory to more general subschemes of projective space.

Definition: A finite subscheme \(\Gamma\) of \(\mathbb{P}^ r\) (over some algebraically closed field) is in linearly general position if for every proper linear subspace \(\Lambda \subset \mathbb{P}^ r\) we have \(\deg \Lambda \cap \Gamma \leq 1 + \dim \Lambda\).

Algebraic interpretation: If we let \(W\) be an \((r+1)\)-dimensional vector space over \(F\), and write \(\mathbb{P}^ r = \mathbb{P}(W)\), then a finite subscheme of \(\mathbb{P}^ r\) corresponds (cf. §1) to a finite-dimensional \(F\)-algebra \(A = {\mathcal O}_ \Gamma (\Gamma)\) and a map from \(W\) to \(A\) whose image includes the identity element. It turns out that the subscheme is in linearly general position iff for every ideal \(I\) of \(A\), the composite map \(W \to A \to A/I\) is either a monomorphism or an epimorphism. – We will be interested here in whether the lemma of Castelnuovo, which says that a (reduced) set of \(r+3\) points in linearly general position in \(\mathbb{P}^ r\) must lie on a rational normal curve, remains valid in the context of schemes. Elementary examples suggest that this might not be the case! The first infinitesimal neighborhood of a point in \(\mathbb{P}^ r\) is a scheme of degree \(r+1\) in linearly general position, and certainly lies on no smooth curves at all if \(r \geq 2\). We shall see similar examples for all \(r\) later on. However, as soon as the degree of \(\Gamma\) is at least \(r+3\), such examples are impossible, and the scheme-theoretic version of Castelnovo’s lemma does hold.

Main result. (Theorem 1): Suppose \(\Gamma\) is a finite subscheme of \(\mathbb{P}^ r\) in linearly general position over an algebraically closed field.

(a) If \(\deg \Gamma \geq r + 3\), then \(\Gamma\) lies on a smooth curve which is unramified at each point in the support of \(\Gamma\).

(b) If \(\deg \Gamma = r + 3\), then \(\Gamma\) lies on a unique (smooth) rational normal curve of degree \(r\).

Corollary. Any two subschemes \(\Gamma\) and \(\Gamma'\) of degree \(r+3\) in linearly general position in \(\mathbb{P}^ r\) are conjugate by an automorphism of \(\mathbb{P}^ r\) provided that they are the same as cycles and their supports contain at most three points.

Corollary. If \(\Gamma\) is a finite subscheme of degree \(\geq r + 3\) in linearly general position in \(\mathbb{P}^ r\) and \(q\) is a general point, then \(\Gamma \cup \{q\}\) is in linearly general position.

Definition: A finite subscheme \(\Gamma\) of \(\mathbb{P}^ r\) (over some algebraically closed field) is in linearly general position if for every proper linear subspace \(\Lambda \subset \mathbb{P}^ r\) we have \(\deg \Lambda \cap \Gamma \leq 1 + \dim \Lambda\).

Algebraic interpretation: If we let \(W\) be an \((r+1)\)-dimensional vector space over \(F\), and write \(\mathbb{P}^ r = \mathbb{P}(W)\), then a finite subscheme of \(\mathbb{P}^ r\) corresponds (cf. §1) to a finite-dimensional \(F\)-algebra \(A = {\mathcal O}_ \Gamma (\Gamma)\) and a map from \(W\) to \(A\) whose image includes the identity element. It turns out that the subscheme is in linearly general position iff for every ideal \(I\) of \(A\), the composite map \(W \to A \to A/I\) is either a monomorphism or an epimorphism. – We will be interested here in whether the lemma of Castelnuovo, which says that a (reduced) set of \(r+3\) points in linearly general position in \(\mathbb{P}^ r\) must lie on a rational normal curve, remains valid in the context of schemes. Elementary examples suggest that this might not be the case! The first infinitesimal neighborhood of a point in \(\mathbb{P}^ r\) is a scheme of degree \(r+1\) in linearly general position, and certainly lies on no smooth curves at all if \(r \geq 2\). We shall see similar examples for all \(r\) later on. However, as soon as the degree of \(\Gamma\) is at least \(r+3\), such examples are impossible, and the scheme-theoretic version of Castelnovo’s lemma does hold.

Main result. (Theorem 1): Suppose \(\Gamma\) is a finite subscheme of \(\mathbb{P}^ r\) in linearly general position over an algebraically closed field.

(a) If \(\deg \Gamma \geq r + 3\), then \(\Gamma\) lies on a smooth curve which is unramified at each point in the support of \(\Gamma\).

(b) If \(\deg \Gamma = r + 3\), then \(\Gamma\) lies on a unique (smooth) rational normal curve of degree \(r\).

Corollary. Any two subschemes \(\Gamma\) and \(\Gamma'\) of degree \(r+3\) in linearly general position in \(\mathbb{P}^ r\) are conjugate by an automorphism of \(\mathbb{P}^ r\) provided that they are the same as cycles and their supports contain at most three points.

Corollary. If \(\Gamma\) is a finite subscheme of degree \(\geq r + 3\) in linearly general position in \(\mathbb{P}^ r\) and \(q\) is a general point, then \(\Gamma \cup \{q\}\) is in linearly general position.