Quadrics through a set of points and their syzygies
Let \(X\) be a nondegenerate projective variety in \(\mathbb{P}^ n = P^ n_ k\), where \(k\) is an algebraically closed field. Let \(I = I(X)\) denote the defining ideal of \(X\) in the polynomial ring \(R : = k[X_ 0, \dots, X_ n]\) and \(A : = R/I\) its homogeneous coordinate ring. The graded \(R\)- module \(A\) has a minimal free resolution \(\mathbb{E} : 0 \to E_ h \to \cdots \to E_ 1 \to R \to A \to 0\) where \(h = hd_ R (A)\) and \(E_ i = \oplus^{\beta_ i}_{j=1} R(-d_{ij})\). Several recent investigations and conjectures relate the numerical invariants of the resolution with the geometrical properties of \(X\). In this paper we are mainly concerned with the ``linear part'' of the resolution. Hence for every \(i = 1, \dots, h\), we let \(a_ i (X) = \dim_ k [\text{Tor}^ R_ i (A, k)]_{i + 1}\) to be the multiplicity of the shift \(i + 1\) in \(E_ i\). Hence we are interested in projective varieties \(X\) which lie on some quadric and we want to study the syzygies of the quadrics passing through \(X\). The main idea coming from M. Green is that a long linear strand in the resolution has a uniform and simple motivation. Following this approach we start by proving that \(a_ i \neq 0\) in the following geometric situations. Either \(X\) is contained on a variety of minimal degree and dimension \(n - i\), or \(X\) is contained in the union of two linear subspaces of \(\mathbb{P}^ n\) of dimension \(k\) and \(r\) where \(k, r < n\) and \(r + k = 2n - i - 1\). A natural question is to what extent the converse of the above result is true. In section 1 we prove that this is the case if \(i = 1\) (trivial) or \(i = n\). -- In section 2 we consider the case \(i = n - 1\) and we restrict ourselves to reduced schemes which are zero dimensional. -- A basic result by M. Green, the so-called strong Castelnuovo lemma, says that for a set \(X\) of distinct points in linearly general position in \(\mathbb{P}^ n\), we have \(a_{n-1} \neq 0\) if and only if the points are on a rational normal curve of \(\mathbb{P}^ n\). Here we complete Green's result by proving that for a set \(X\) of distinct points in \(\mathbb{P}^ n\), such that \(n - 1\) are never on a linear subspace of dimension \(n - 3\), we have \(a_{n-1} \neq 0\) if and only if either the points are on a rational normal curve of \(\mathbb{P}^ n\) or \(X \subseteq \mathbb{P}^ k \cup \mathbb{P}^ r\) with \(k + r = n\). We conjecture that this result should be true in complete generality. However even this weaker form of the conjecture, unables us to solve one of the problems which motivated our present research. Namely in section 4 we apply the above theorem to describe in a concrete geometric way the open set where the minimal resolution conjecture holds for a given set of \(n + 4\) points spanning \(\mathbb{P}^ n\). -- In the case \(i = n - 2\) we remark that the analogue of the strong Castelnuovo lemma does not hold. For example 12 general points in \(\mathbb{P}^ 7\) are not on a rational normal scroll of dimension 2 but have \(a_ 5 = 4 \neq 0\). In section 3 we can prove that they are on a threefold of minimal degree as a consequence of a general result which in particular asserts that if \(n \geq 3\) and \(p\) is an integer, \(1 \leq p \leq n - 2\), then every set of \(2n + 1 - p\) points in linearly general position lies on a rational normal scroll of dimension \(n - p - 1\). This result extends a classical theorem of Bertini for \(n + 3\) points in \(\mathbb{P}^ n\).
scientific article; zbMATH DE number 737160