Some examples of Gorenstein liaison in codimension three.

*(English)*Zbl 1076.14065Liaison has shown to be a powerful tool for studying schemes of codimension 2 in any projective space \(\mathbb P^n_K, K\) an algebraically closed field. This is done by complete intersection linkage. It seems that complete intersection liaison in higher codimension is too fine in order to give analogous results in higher codimension. Thus attention has been focused on Gorenstein liaison as a suitable generalization in higher codimension. The purpose of this paper is to give some examples of Gorenstein liaison for points in \(\mathbb P^3_K\) and for curves in \(\mathbb P^4_K\) which indicate that even under Gorenstein liaison the situation in codimension \(\geq 3\) may be more complicated than expected.

To be more precise: For points in \(\mathbb P^3_K\) it is shown that any set of \(n\) points in general position in a plane or on a nonsingular quadric is in the Gorenstein liaison class of a complete intersection (glicci). On a nonsingular cubic surface any set of \(n\) points in general position is “glicci” by ascending and descending biliaisons and simple liaisons. For a set of \(n\) points in \(\mathbb P^3_K\) it is shown that it is “glicci” for \(n \leq 19.\) This is open for \(n = 20.\) So this is a candidate for a possible counterexample to the problem whether all ACM schemes are “glicci”.

Then it is shown that any general arithmetically Cohen-Macaulay (ACM) curve of degree \(\leq 9\) resp. degree \(10\) and genus \(6\) is “glicci”. Moreover, in \(\mathbb P^4_K\) the author studies ACM curves of degree 20 and genus 26. He shows that a general curve in the irreducible component of the Hilbert scheme containing the determinantal curves cannot be obtained by ascending Gorenstein biliaison from a line. Therefore, it is a candidate for an example of an ACM curve that is not “glicci”.

Finally the author considers curves in \(\mathbb P^4_K\) with Hartshorne-Rao module \(K(s)\), \(s \in \mathbb Z.\) It is called minimal whenever \(s=0.\) There are minimal curves in any degree \(\geq 2.\) Then he gives examples that suggest that there are curves in the biliaison class of two skew lines that cannot be reached by ascending Gorenstein biliaison from a minimal curve, and that there are other with the Hartshorne-Rao module \(K(s)\) that are not in the liaison class of two skew lines.

With his considerations the author directs further research towards a better understanding of “glicci”.

To be more precise: For points in \(\mathbb P^3_K\) it is shown that any set of \(n\) points in general position in a plane or on a nonsingular quadric is in the Gorenstein liaison class of a complete intersection (glicci). On a nonsingular cubic surface any set of \(n\) points in general position is “glicci” by ascending and descending biliaisons and simple liaisons. For a set of \(n\) points in \(\mathbb P^3_K\) it is shown that it is “glicci” for \(n \leq 19.\) This is open for \(n = 20.\) So this is a candidate for a possible counterexample to the problem whether all ACM schemes are “glicci”.

Then it is shown that any general arithmetically Cohen-Macaulay (ACM) curve of degree \(\leq 9\) resp. degree \(10\) and genus \(6\) is “glicci”. Moreover, in \(\mathbb P^4_K\) the author studies ACM curves of degree 20 and genus 26. He shows that a general curve in the irreducible component of the Hilbert scheme containing the determinantal curves cannot be obtained by ascending Gorenstein biliaison from a line. Therefore, it is a candidate for an example of an ACM curve that is not “glicci”.

Finally the author considers curves in \(\mathbb P^4_K\) with Hartshorne-Rao module \(K(s)\), \(s \in \mathbb Z.\) It is called minimal whenever \(s=0.\) There are minimal curves in any degree \(\geq 2.\) Then he gives examples that suggest that there are curves in the biliaison class of two skew lines that cannot be reached by ascending Gorenstein biliaison from a minimal curve, and that there are other with the Hartshorne-Rao module \(K(s)\) that are not in the liaison class of two skew lines.

With his considerations the author directs further research towards a better understanding of “glicci”.

Reviewer: Peter Schenzel (Halle)