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# is a tensorcartesian square of a group scheme with $\mathbb{G}_a^n$ fibres reduced?

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Let $G$ be a group scheme over $S$ where $S$ is a reduced scheme of finite type over a field $k$ of characteristic 0, and let every fibre $G_s$ over a closed point of $S$ be isomorphic to $\mathbb{G}_a^n$ for some $n$ that varies with $s$.

Is it true then that $G$ is reduced?

Suppose that $G$ is reduced. Is it true that $G \otimes_S times_S G$ is reduced?

(This question is a continuation of this one; the motivation comes from this question)

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# is a tensorsquareofa group scheme with $\mathbb{G}_a^n$ fibres reduced?

(This question is a continuation of this one)

Let $G$ be a group scheme over $S$ where $S$ is a reduced scheme of finite type over a field $k$ of characteristic 0, and let every fibre $G_s$ over a closed point of $S$ be isomorphic to $\mathbb{G}_a^n$ for some $n$ that varies with $s$.

Is it true then that $G$ is reduced?

Is it true that $G \otimes_S G$ reduced?

(This question is a continuation of this one; the motivation comes from this question)

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