Triads and families of space curves
Many interesting properties of curves \(C\) in the projective space \({\mathbb P}^3\) can be read in the Hartshorne-Rao module \(\bigoplus^nH^1({\mathcal I}_C(n))\), which contains informations on the extrinsic geometry of \(C\). Unfortunately, when \(C\) is replaced by a flat family of curves \({\mathcal C}\) defined over the local ring \(A\), the naive definition of the Hartshorne-Rao module is useless because it does not commute with base change. The authors propose a different approach, namely they associate to any curve a triad, i.e. a complex of free modules \(L_\bullet= L_1\to L_0\to L_{-1}\) whose middle cohomology is the module \(\bigoplus^nH^1({\mathcal I}_C(n))\). This triad is obtained from sheaf resolutions \(0\to {\mathcal P}\to {\mathcal N}\to {\mathcal I}_C\to 0\) and \(0\to {\mathcal N}\to {\mathcal L}_1 \to {\mathcal L}_0\to {\mathcal L}_{-1}\to 0\), such that \({\mathcal P}\) and the \({\mathcal L}_i\)'s are sums of line bundles. When \(C\) is replaced by a family in \({\mathbb P}^3_A\), \(A\) a local ring, then the triad behaves properly with respect to the base change. The authors introduce a general theory of triads and triadic sheaves and they prove that triads associated to families are good surrogates for the Hartshorne-Rao modules of curves in many senses. For instance, they show that two families of curves are in the same biliaison classes if and only if the corresponding triads are pseudo-isomorphic (up to twisting). Several examples of concrete computation of triads associated to families of curves are provided.
scientific article; zbMATH DE number 1384375