On the Extensions of Group Schemes Deforming $\mathbf{G}_a$ to $\mathbf{G}_m$

Abstract

For given group schemes $\mathcal{G}^{(\lambda_i)}\ (i=1,2,\ldots)$ deforming the additive group scheme $\mathbf{G}_a$ to the multiplicative group scheme $\mathbf{G}_m$, T. Sekiguchi and N. Suwa constructed extensions: $$0\rightarrow\mathcal{G}^{(\lambda_2)}\rightarrow\mathcal{E}^{(\lambda_1,\lambda_2)}\rightarrow\mathcal{G}^{(\lambda_1)}\rightarrow0,\ \ldots,\ 0\rightarrow\mathcal{G}^{(\lambda_n)}\rightarrow\mathcal{E}^{(\lambda_1,\ldots,\lambda_n)}\rightarrow\mathcal{E}^{(\lambda_1,\ldots,\lambda_{n-1})}\rightarrow0,\ldots$$ inductively, by calculating the group of extensions $\mathrm{Ext}^1(\mathcal{E}^{(\lambda_1,\ldots,\lambda_{n-1})},\mathcal{G}^{(\lambda_n)})$. Here we treat the group $\mathrm{Ext}^1(\mathcal{G}^{(\lambda_0)},\mathcal{E}^{(\lambda_1,\ldots,\lambda_n)})$ of extensions in the case of $n=2,\ 3$. The case of $n=2$ was studied by D. Horikawa.