Free resolutions of the defining ideal of certain rational surfaces (Q1914847)

The surfaces \(X\subseteq \mathbb{P}^5_k\) which are considered in this paper are obtained this way: Let \(Y\) be the blow-up of \(\mathbb{P}^2 = \mathbb{P}^2_k\) (where \(k\) is an algebraically closed field) at \(n= t^2\) distinct points \(P_1, \dots, P_n\) which are the complete intersection of two curves of degree \(t\), and then consider the immersion \(\varphi:Y\to\mathbb{P}^5\) defined by the linear system of the curves (in \(\mathbb{P}^2)\) of degree \(t+1\) containing \(P_1, \dots, P_n\). The minimal free resolution of the homogeneous ideal \(I_X\) (in the ring \(S= k[x_0, \dots, x_5])\) is determined by considering the rational normal cubic threefold \(T \supset X\) defined by the \(2 \times 2\) minors of the matrix \({X_0 X_2X_4 \choose X_2 X_3X_5}\). We have \(I_T \subset I_X\), and it is shown (proposition 2.2) that the resolution of \(I_X\) is given by the direct sum of the resolutions (as \(S\)-modules) of \(I_T\) (which is known) and \(I_X/I_T\) (which is computed).