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Finite projective schemes in linearly general position. (English) Zbl 0804.14002
If $$X$$ 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.

##### MSC:
 14A15 Schemes and morphisms 14N05 Projective techniques in algebraic geometry 14C99 Cycles and subschemes