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Using the possibility of computationally determining points on a finite cover of a unirational variety over a finite field, we determine all possibilities for direct Gorenstein linkages between general sets of points in P3 over an algebraically closed field of characteristic 0. As a consequence, we show that a general set of d points is glicci (that is, in the Gorenstein linkage class of a complete intersection) if d ≤33 or d = 37, 38. Computer algebra plays an essential role in the proof. The case of 20 points had been an outstanding problem in the area for a dozen years [8]. For Rob Lazarsfeld on the occasion of his 60th birthday

Using the possibility of computationally determining points on a finite cover of a unirational variety over a finite field, we determine all possibilities for direct Gorenstein linkages between general sets of points in P3 over an algebraically closed field of characteristic 0. As a consequence, we show that a general set of d points is glicci (that is, in the Gorenstein linkage class of a complete intersection) if d ≤33 or d = 37, 38. Computer algebra plays an essential role in the proof. The case of 20 points had been an outstanding problem in the area for a dozen years [8]. For Rob Lazarsfeld on the occasion of his 60th birthday

We describe some recent work concerning Gorenstein liaison of codimension twosubschemes of a projective variety. Applications make use of the algebraictheory of maximal Cohen-Macaulay modules, which we review in an Appendix.

We study Gorenstein liaison of codimension two subschemes of anarithmetically Gorenstein scheme X. Our main result is a criterion for two suchsubschemes to be in the same Gorenstein liaison class, in terms of the categoryof ACM sheaves on X. As a consequence we obtain a criterion for X to have theproperty that every codimension 2 arithmetically Cohen-Macaulay subscheme is inthe Gorenstein liaison class of a complete intersection. Using these tools weprove that every arithmetically Gorenstein subscheme of $\mathbb{P}^n$ is inthe Gorenstein liaison class of a complete intesection and we are able tocharacterize the Gorenstein liaison classes of curves on a nonsingular quadricthreefold in $\mathbb{P}^4$.

Let $X$ be a normal arithmetically Gorenstein scheme in ${\mathbb P}^n$. Wegive a criterion for all codimension two ACM subschemes of $X$ to be in thesame Gorenstein biliaison class on $X$, in terms of the category of ACM sheaveson $X$. These are sheaves that correspond to the graded maximal Cohen--Macaulaymodules on the homogeneous coordinate ring of $X$. Using known results on MCMmodules, we are able to determine the Gorenstein biliaison classes ofcodimension two subschemes of certain varieties, including the nonsingularquadric surface in ${\mathbb P}^3$, and the cone over it in ${\mathbb P}^4$. Asan application we obtain a new proof of some theorems of Lesperance aboutcurves in ${\mathbb P}^4$, and answer some questions be raised.

Let $X$ be an integral projective scheme satisfying the condition $S_3$ ofSerre and $H^1({\mathcal O}_X(n)) = 0$ for all $n \in {\mathbb Z}$. Wegeneralize Rao's theorem by showing that biliaison equivalence classes ofcodimension two subschemes without embedded components are in one-to-onecorrespondence with pseudo-isomorphism classes of coherent sheaves on $X$satisfying certain depth conditions. We give a new proof and generalization of Strano's strengthening of theLazarsfeld--Rao property, showing that if a codimension two subscheme is notminimal in its biliaison class, then it admits a strictly descending elementarybiliaison. For a three-dimensional arithmetically Gorenstein scheme $X$, we show thatbiliaison equivalence classes of curves are in one-to-one correspondence withtriples $(M,P,\alpha)$, up to shift, where $M$ is the Rao module, $P$ is amaximal Cohen--Macaulay module on the homogeneous coordinate ring of $X$, and$\alpha: P^{\vee} \to M^* \to 0$ is a surjective map of the duals.

We extend the theory of generalized divisors so as to work on any scheme $X$satisfying the condition $S_2$ of Serre. We define a generalized notion ofGorenstein biliaison for schemes in projective space. With this we give a newproof in a stronger form of the theorem of Gaeta, that standard determinantalschemes are in the Gorenstein biliaison class of a complete intersection. We also show, for schemes of codimension three in ${\mathbb P}^n$, that therelation of Gorenstein biliaison is equivalent to the relation of even strictGorenstein liaison.