Minimal free resolution for points on surfaces
Let \(X \subset \mathbb{P}^n\) be a subscheme and let \[ F_\bullet: 0 \longrightarrow F_{n+1} {\longrightarrow} F_n \longrightarrow \cdots \longrightarrow F_1 {\longrightarrow} F_0 {\longrightarrow} R/I(X) \longrightarrow 0 \] be its minimal graded free resolution. The graded Betti numbers \(b_{ij}(X)\) are defined as \[ F_i=\bigoplus_{j \in \mathbb{Z}}R(-i-j)^{b_{i,j}(X)}. \] Let \(Z \subset X\) be a set of distinct points on \(X\) with homogeneous ideal \(I(X)\). A very long-standing problem in algebraic geometry and commutative algebra is to determine the Hilbert function of \(k[x_0, \dots, x_n]/I(Z)\). Then a much more subtle question is to understand all of the syzygies of \(I(Z)\), i.e. to find the minimal graded free resolution of \(k[x_0, \dots, x_n]/I(Z)\) or, at least, the graded Betti numbers \(b_{i,j}(Z)\) of \(Z\). In 1993, \textit{D. Lorenzini} [J. Algebra 156, No. 1, 5--35 (1993; Zbl 0811.13008)] stated the so-called Minimal Resolution Conjecture (MRC) that, grosso modo, says that a general set \(Z\) of \(s\) distinct points in \(\mathbb{P}^n\) has no ghost terms, i.e. \(b_{i+1,j}(Z) \cdot b_{i,j+1}(Z) =0\) for all \(i,j\). The conjecture holds for \(n\leq 4\) and for large values of \(s\) for any \(n\), but it fails for all \(n\geq 6\), \(n\not= 9\). In 1998, \textit{M. Mustaţă} [Matematiche 53, Suppl., 53--81 (1998; Zbl 0943.13010)] introduced a generalized version of MRC for distinct points in arbitrary projective varieties. Let \(\mathrm{Hilb}^s(X)\) be the Hilbert scheme parameterizing \(0-\)dimensional subschemes of \(X\) of length \(s\) and let \(H^s_0\) be the irreducible component whose general points correspond to a set \(Z\) of \(s\) distinct points on \(X\). \textit{Mustaţǎ's conjecture}. Let \(X \subset \mathbb{P}^n\) be a projective variety, let \(P_X(t)\) be its Hilbert polynomial and \(m=\mathrm{reg}(X)\). Let \(s\) be an integer such that \(P_X(r-1) \leq s < P_X(r )\) for some \(r\geq m+1\). Then, there is a non-empty open subset \(U^s_0 \subset H^s_0 \subset \mathrm{Hilb}^s(X)\) such that for any \([Z]\in U_0^s\) we have \[ b_{i+1,r-1}(Z) \cdot b_{i,r}(Z) =0 \quad { \text{for }}\quad i=1, \dots, n-1. \] In this paper the authors state a refined version of Mustaţǎ's conjecture and predict the existence of a non-empty open subset \(U \subset \mathrm{Hilb}^s(X)\) such that any \([Z] \in U\) has a minimal graded free resolution without ghost terms: Conjecture 1.2. Let \(X \subset \mathbb{P}^n\) be a projective variety, let \(P_X(t)\) be its Hilbert polynomial and \(m=\mathrm{reg}(X)\). Let \(s\) be an integer such that \(P_X(r-1) \leq s < P_X(r )\) for some \(r\geq m+1\). Then, there is an irreducible component \(H^s \subset \mathrm{Hilb}^s(X)\) and a non-empty open subset \(U^s \subset H^s \subset \mathrm{Hilb}^s(X)\) such that for any \([Z]\in U^s\) we have \[ b_{i+1,r-1}(Z) \cdot b_{i,r}(Z) =0 \quad {\text{for} } \quad i=1, \dots, n-1. \] In the paper the authors address both conjectures for surfaces in \(\mathbb{P}^3\) and they prove: (1) for any integer \(d\geq 2\) and for any \(s \geq {d+3 \choose 3}-1\), there exists a \({d+2 \choose 2}-\)dimensional family of irreducible generically smooth surfaces \(X \subset \mathbb{P}^3\) of degree \(d\) satisfying Conjecture 1.2; (2) a general set of \(s \geq 19\) distinct points on any smooth cubic in \(\mathbb{P}^3\) satisfies Mustaţǎ's conjecture.
scientific article; zbMATH DE number 6104541
scientific article