The role of the cotangent bundle in resolving ideals of fat points in the plane (Q958119)

The paper concerns with minimal free graded resolutions of fat point ideals in \(\mathbb{P}^2\). Given general points \(P_1,\dots, P_n \in \mathbb{P}^2\) and nonnegative integers \(m_1,\dots,m_n\), the fat point ideal \(I(Z)\) is defined as the ideal \(I(P_1)^{m_1}\cap \cdots \cap I(P_n)^{m_n}\) of \(R=K[x_0, x_1, x_2]\), where \(K\) is any algebraically closed field and where \(I(P_i)\) is the ideal generated by all forms that vanish at \(P_i\). In order to understand better the geometry of \(Z\) as a subscheme of \(\mathbb{P}^2\), the first approach concernsthe study of the K-vector space of curves of given degree k containing Z, that is, having singularities of multiplicity at least \(m_1,\dots,m_n\) at the given points \(P_1,\dots, P_n\). This is equivalent to determine the dimension of the homogeneous component \(I(Z)_k\) of \(I(Z)\). To go deeper into the geometry of a fat point scheme, the next step consists in understanding the relations among the curves containing \(Z\), that is, determining the minimal free graded resolution \(0\to M_1 \to M_0 \to I(Z)\to 0\) of \(I(Z)\), where \(M_0\) and \(M_1\) are free \(R-\)modules of the form \(M_0 = \bigoplus_k R^{t_k}[-k]\) and \(M_1 = \bigoplus_k R^{s_k}[-k]\). If the Hilbert function \(h_Z\) is known and if the graded Betti numbers \(t_k\) are known, then the values of \(s_k\) are easy to determine from the exact sequence above. Hence the attention is in the graded Betti numbers \(t_k\). It is proved that \(t_k\) is the dimension of the cokernel of the map \(\mu_{k-1}(Z) : I(Z)_{k-1}\otimes R_1 \to I(Z)_k\), where \(R_1\) denotes the \(K-\)vector space spanned in \(R\) by linear forms and \(\mu_{k-1}\) is the map induced by multiplication of elements of \(I(Z)_{k-1}\) by linear forms. The authors propose a reflection about the geometric obstacles to the rank maximality of the maps \(\mu_k\). Let \(\Omega\) be the cotangent bundle of \(\mathbb{P}^2\), and by \(p:X \to \mathbb{P}^2\) the blow up at the points \(P_i\). The problem of determining the rank of the maps \(\mu_k(Z)\) is equivalent to two different postulation problems for \(Z\), one in \(\mathbb{P}^2\) and the other in \(X\): determine, for each \(k\), the rank of the restriction map (a) \(\rho_k=\rho_k(Z):H^0(\Omega(k+1))\to H^0(\Omega(k+1)_{|Z})\); or (b) \(\nu_k=\nu_k(Z):H^0(p^*\Omega(k+1))\to H^0(p^*\Omega(k+1)_{|p^{-1}Z})\). The authors show that the point of view (a) gives some information about the failure of this rank maximality due to superfluous conditions imposed by \(Z\) to the restriction of to some curves; but in fact this is not enough, and the right point of view is (b), since it is then possible to take into account the splitting of \(p^*\) on the normalization of the appropriate rational curves, and this allows to count properly the superfluous conditions imposed by \(Z\) to the restriction of to each curve. By studying several examples and proving certain results the paper arrives at two conjectures about the failure of the rank maximality of \(\mu_k\), one when \(\mu_k\) is expected to be surjective and the other when injectivity is expected. Notice that a similar line of thought leads to the Segre-Harbourne-Gimigliano-Hirschowitz Conjecture. The use of the cotangent bundle in problems concerning the generation of homogeneous ideals of subschemes of a projective space was introduced by A. Hirschowitz, and used for the first time for curves in \(\mathbb{P}^3\).