Patrick Polo
Sorbonne université, Paris
January 30, 2025
Rank $2$ vector bundles over the projective line and catalecticant matrices: Let $k$ be a field. It is well-known that over $X = \mathbb{P}^1_k$ every vector bundle $\mathcal{E}$ is a direct sum of line bundles. On the other hand, if $\mathcal{E}$ is generated by global sections and of rank $r\geq 2$, it contains a trivial bundle of rank $r-1$ and the quotient is the line bundle $\det(\mathcal{E})$. Consider the simplest case $r=2$ and fix a positive integer $n$. The extensions $\mathcal{E}$ of $\mathcal{O}_X(n)$ by $\mathcal{O}_X$ are parametrized by the vector space $V_n = H^1(X,\mathcal{O}_X(-n))$ which, by Serre duality, is dual to $H^0(X,\mathcal{O}_X(n-2))$. The trivial extension corresponds to the null vector. So, assuming $n\geq 2$ and discarding the trivial extension, we obtain a partition of $\mathbb{P}(V_n)(k)$ %by subsets corresponding to the isomorphism type of $\mathcal{E}$, namely $\mathcal{O}_X(j) \oplus \mathcal{O}_X(n-j)$ for $1\leq j \leq n-j$. One may wonder what is this partition. It turns out that this a stratification by locally closed subschemes, which have been studied in invariant theory: their closures are defined by the vanishing of minors of a certain catalecticant matrix. The proof relies on known results, some of them going back to F. S. Macaulay in 1916.
Sorbonne université, Paris
January 30, 2025
Rank $2$ vector bundles over the projective line and catalecticant matrices: Let $k$ be a field. It is well-known that over $X = \mathbb{P}^1_k$ every vector bundle $\mathcal{E}$ is a direct sum of line bundles. On the other hand, if $\mathcal{E}$ is generated by global sections and of rank $r\geq 2$, it contains a trivial bundle of rank $r-1$ and the quotient is the line bundle $\det(\mathcal{E})$. Consider the simplest case $r=2$ and fix a positive integer $n$. The extensions $\mathcal{E}$ of $\mathcal{O}_X(n)$ by $\mathcal{O}_X$ are parametrized by the vector space $V_n = H^1(X,\mathcal{O}_X(-n))$ which, by Serre duality, is dual to $H^0(X,\mathcal{O}_X(n-2))$. The trivial extension corresponds to the null vector. So, assuming $n\geq 2$ and discarding the trivial extension, we obtain a partition of $\mathbb{P}(V_n)(k)$ %by subsets corresponding to the isomorphism type of $\mathcal{E}$, namely $\mathcal{O}_X(j) \oplus \mathcal{O}_X(n-j)$ for $1\leq j \leq n-j$. One may wonder what is this partition. It turns out that this a stratification by locally closed subschemes, which have been studied in invariant theory: their closures are defined by the vanishing of minors of a certain catalecticant matrix. The proof relies on known results, some of them going back to F. S. Macaulay in 1916.