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Let $k$ be an algebraically closed field of characteristic zero. Let $X$ be a projective scheme over $k$. We can talk about short exact sequences of coherent sheaves on $X$. Suppose we have a family of SES's parametrized by $T$, a scheme of finite type over $k$, that is a short exact sequence of coherent sheaves on $X\times_kT$ $$0\to E\to F\to G\to0$$ such that

  1. $E,F,G$ are coherent sheaves on $X\times_kT$, flat over $T$;
  2. for each closed point $t\in T$, the fibre over $X_t:=X\times_k\mathrm{Spec}(\kappa(t))$ is exact (As commented, this condition can be derived from flatness and exactness of $0\to E\to F\to G\to0$. ) $$0\to E|_{X_t}\to F|_{X_t}\to G|_{X_t}\to0\qquad(*)$$

My question is about the ''splitting locus'' $$\{t\in T:t\textrm{ is a closed point, }(*) \textrm{ is splitting}\}. $$ Now it is only defined as a subset of closed points. I wonder if this can be enhanced as a subscheme.

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    $\begingroup$ Your condition 2. follows from 1. $\endgroup$
    – Sasha
    Commented Apr 17, 2022 at 17:32
  • $\begingroup$ I wonder if the natural map $\mathrm{Ext}^1_{X \times T}(G, E) \to H^0(\mathcal{E}xt^1_{X \times T}(G,E))$ can be used here. That should at least detect some kind of local splitting on $X \times T$, if not fiberwise splitting (maybe we need $R^1\pi_{2*}$?). $\endgroup$
    – Jake Levinson
    Commented Apr 17, 2022 at 22:19

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