MathOverflow

I'm not an expert on Bhargava's work Higher composition laws I https://annals.math.princeton.edu/wp-content/uploads/annals-v159-n1-p03.pdf. But the following "counterexample" to Theorem 11 of the paper bothers me. Following the same notation as in the...

Let $k$ be an algebraically closed field. Consider a smooth projective variety $X/k$ and a smooth quasi-projective variety $Y/k$. Then by a theorem of Grothendieck, the Hom functor $\underline{\text{Hom}}_k(X, Y)$ is represented by a scheme.
Do we know how to study the reducedness of this scheme? I understand that one can use the general deformation theory to study the smoothness at some point $[\varphi]$, but that is a harder question and reducedness is enough for me....

[Crossposted at math.stackexchange]
Let $\mathcal{F} = \{\{x_1, x_2\} : 1 \le x_1 \lt x_2 \le n \}$, $n \ge 2$, and let $\mathcal{G} = \{G_1, \ldots, G_n\}$ be a partition of $\mathcal{F}$ in $n$ parts. Suppose WLOG $|G_1| \ge |G_2| \ge \cdots \ge |G_{n-1}| \ge |G_{n}|$....

Let's say that a real number has a simple trigonometric representation, if it can be represented as a product of zero or more rational powers of positive integers and zero or more (positive or negative) integer powers of $\sin(\cdot)$ at rational multiples...