On the invariants of pencils of binary quintics.
A pencil of quintic binary forms is a \(2\)-dimensional subspace of \(\mathbf{S} _{5},\) where \(\mathbf{S}_{5}\) is the \({\text{SL}_{2}}\)-module \( H^{0}\left( \mathcal{O}_{\mathbb{P}^1}\left( 5\right) \right) .\) This paper is a study of the invariants of the pencils of binary quintic forms under the natural action of \({\text{SL}_{2}}.\) As the Grassmannian \( \mathbf{G}_{2}\left( \mathbf{S}_{5}\right) \) is birationally equivalent and \( {\text{SL}_{2}}\)-equivariant to \(\mathbb{P}\left( \mathbf{S}_{8}\right) \) we consider the algebra of invariants of an octic form. Furthermore, the invariants of an octic form corresponds to a Gorenstein algebra of codimension 3. The main result is as follows. For \(B\) the \textit{algèbre du cône} of \(\mathbf{G}_{2}\left( \mathbf{S}_{5}\right) \) and \(R=\mathbb{C}\left[ x_{1},x_{2},x_{3},x_{3}^{\prime },x_{4},x_{5},x_{5}^{\prime },x_{6},x_{7} \right] \) (where the degree of each indeterminate is its index), the graded algebra \(B^{\text{SL}_{2}}\) is isomorphic to \(R/I,\) where \(I\) is an ideal of Pfaffians of order \(4\) of a certain alternating \(5\times 5 \) matrix. Furthermore, \[ 0\rightarrow R\left( -30\right) \rightarrow \bigoplus_{d=16}^{20}R\left( -d\right) \rightarrow \bigoplus_{d=10}^{14}R\left( -d\right) \rightarrow R\rightarrow 0 \] is the minimal projective resolution for \(B^{\text{SL}_{2}}.\)
scientific article; zbMATH DE number 2123558
scientific article