On the minimal free resolution of general \(k\)-gonal curves
This paper studies minimal free resolutions of ideal sheaves of projective curves \(C \subset {\mathbb P}^n\). Let \[ 0 \rightarrow E_n \rightarrow\dots\rightarrow E_1 \rightarrow {\mathcal I}_C \rightarrow 0 \] be a minimal resolution of the ideal sheaf of \(C\); we say that \(C\) has property \(N_0\) if it is a.C.M. and that it has property \(N_p\), \(p\geq 1\), if \(N_0\) holds and each \(E_i \equiv \sum {\mathcal O}_{{\mathbb P}^n}(-i-1)\). In a previous paper [Pac. J. Math. 172, No. 2, 315-319 (1996; Zbl 0848.14007)], \textit{E. Ballico} has given a bound \(G(p,n)\) such that for each \(g \leq G(p,n)\), the general linearly normal non-special curve \(C \subset {\mathbb P}^n\), with genus \(g\) and degree \(g+n\) satisfies property \(N_p\). The numbers are such that we get property \(N_p\) for \(\deg(C)g + \sqrt {2(p+1)g}\). In this paper the methods used to get the previous result are generalized in order to get the same statement also for \(k\)-gonal curves, where \(k > n \geq p\), \(n \geq 3\). Moreover, results are given about the linearity of the resolution also in the case of non-linearly normal, non-special projective curves.
minimal free resolutions of ideal sheaves
scientific article; zbMATH DE number 1051526
scientific article