Minimal Links and a Result of Gaeta

If V is an equidimensional codimension c subscheme of an n-dimensional projective space, and V is linked to V ′ by a complete intersection X, then we say that V is minimally linked to V ′ if X is a codimension c complete intersection of smallest degree containing V . Gaeta showed that if V is any arithmetically Cohen-Macaulay (ACM) subscheme of codimension two then there is a finite sequence of minimal links beginning with V and arriving at a complete intersection. Gaeta’s result leads to two natural questions. First, in the codimension two, non-ACM case, there is no hope of linking V to a complete intersection. Nevertheless, an analogous question can be posed by replacing the target “complete intersection” with “minimal element of the even liaison class” and asking if the corresponding statement is true. Despite a (deceptively) suggestive recent result of Hartshorne, who generalized a theorem of Strano, we give a negative answer to this question with a class of counterexamples for codimension two subschemes of projective n-space. On the other hand, we show that there are even liaison classes of non-ACM curves in projective 3-space for which every element admits a sequence of minimal links leading to a minimal element of the even liaison class. (In fact, in the classes in question, even and odd liaison coincide.) The second natural question arising from Gaeta’s theorem concerns higher codimension. In earlier work with Huneke and Ulrich, we showed that the statement of Gaeta’s theorem as quoted above is false if “codimension two” is replaced by “codimension ≥ 3,” at least for subschemes that admit a sequence of links to a complete intersection (i.e. licci subschemes). Here we show that in the non-ACM situation, the analogous statement is also false. However, one can refine the question for codimension 3 licci subschemes by asking if it is true for arithmetically Gorenstein, codimension 3 subschemes, which Watanabe showed to be licci. Watanabe’s work was extended by Hartshorne, who showed that the general such subscheme of fixed Hilbert function and of dimension 1 can be obtained by a sequence of strictly ascending biliaisons from a linear complete intersection. (Hartshorne, Sabadini and Schlesinger proved the analogous result for arithmetically Gorenstein zerodimensional schemes.) In contrast to the previous results, here we show that in fact for any codimension 3 arithmetically Gorenstein subscheme, in any projective space, a sequence of minimal links does lead to a complete intersection, giving a different extension of Watanabe’s result. Furthermore, we extend Hartshorne’s result by removing the generality assumption as well as the dimension assumption.