We describe recent joint work with U. Nagel, as well as slightly less 
recent (2007) work with C. Huneke, U. Nagel and B. Ulrich.

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 {\em minimally 
linked} to $V'$ if $X$ is a codimension $c$ complete intersection of 
smallest degree containing $V$.  F. 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 ``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 R. Hartshorne, who generalized a result of R. 
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 {\em 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 the paper with Huneke, Nagel and Ulrich we 
show that the statement of Gaeta's theorem as quoted above is false 
if ``codimension two" is replaced by ``codimension $\geq 3$," at 
least for subschemes that  admit a sequence of links to a complete 
intersection (i.e. licci subschemes).  In the work with Nagel, 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 {\em arithmetically 
Gorenstein, codimension 3} subschemes, which J. Watanabe showed to be 
licci.  Watanabe's work was extended by Hartshorne, who showed that 
that the {\em 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 zero-dimensional 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 {\em 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.