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Usually in the moduli space of semistable sheaves, two semistable sheaves correspond to one point if and only if they are S-equvialent, i.e. the graded objects associated to their Jordan-Holder filtration are isomorphic.

I am now interested in distinguishing S-equivalent but non-isomorphic semistable sheaves. The first question is

Are there any researches on this moduli problem?

My first idea is that, we can try the classification of non-split (length 2) Jordan-Holder filtrations, which is a subclass of semistable but not polystable sheaves. The associated graded would be a point in the product of two moduli spaces of stable sheaves.

More concretely, if $0=E_{\leq0}\subsetneq E_{\leq1}\subsetneq E_{\leq 2}=E$ is a Jordan-Holder filtration of length 2. Then $(E_1,E_2)$ is a point in $M_1^s\times M_2^s$, where $E_i:=E_{\leq i}/E_{\leq i-1}$ and $M_i^s$ is the moduli space of stable sheaves. If $M^{JH}_{l=2}$ exists as the moduli space of length 2 non-splitting JH filtrations, then there is a morphism $$M^{JH}_{l=2}\to M_1^s\times M_2^s. $$ The fibre of this morphism at $(E_1,E_2)$ should be $\mathbb{P}(\mathrm{Ext}^1(E_2,E_1))$. Then the second question is

In what sense can we glue $\mathbb{P}(\mathrm{Ext}^1(E_2,E_1))$ over $M_1^s\times M_2^s$ or over some (locally closed) subscheme?

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  1. Sometimes you can perturb polarization (or more generally, stability condition) so that the slopes of $E_1$ and $E_2$ become different. Then the moduli space for the slightly perturbed stability is the blowup of the original moduli space along $M_1^s \times M_2^s$, and the exceptional divisor is the projectivized $\mathrm{Ext}$-space. This phenomenon is known and the name wall-crossing.

  2. To make sense of the projectivized $\mathrm{Ext}$-space in general you can consider the universal sheaves $\mathcal{E}_1$ and $\mathcal{E}_2$ on $M_1^s \times X$ and $M_2^s \times X$ respectively (if they exist), and consider the first derived pushforward $$ \mathcal{V} := R^1 p_{1,2}{}_*(p_{1,3}^*\mathcal{E}_1 \otimes p_{2,3}^*\mathcal{E}_2^\vee), $$ where $p_{i,j}$ are the projections of $M_1^s \times M_2^s \times X$ to the products of the respective factors. Then just consider the projectivization of $\mathcal{V}$.

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  • $\begingroup$ The $\Bbbk$-points of the projectivized Ext-sapce are extensions $0\to E_1\to E\to E_2\to0$ over $X_\Bbbk$ that extend to $0\to \mathcal{E}_1\to\mathcal{E}\to\mathcal{E}_2\to0$ over $X\times M_1^s\times M_2^s$, aren't they? I do not think all extensions in $\mathrm{Ext}^1(E_2,E_1)$ are extendable. We need some base change assumption. $\endgroup$
    – Yikun Qiao
    Commented May 30, 2023 at 13:01
  • $\begingroup$ No, extendability constraint is irrelevant. $\endgroup$
    – Sasha
    Commented May 30, 2023 at 13:33
  • $\begingroup$ Are there any references on this? I still can not get rid of this extendability. $\endgroup$
    – Yikun Qiao
    Commented May 31, 2023 at 5:40
  • $\begingroup$ In a slightly different GIT context there is a nice paper Thaddeus, Michael Geometric invariant theory and flips. J. Amer. Math. Soc. 9 (1996), no. 3, 691–723. In the context of coherent sheaves try googling for "wall-crossing" and "stability conditions". $\endgroup$
    – Sasha
    Commented May 31, 2023 at 5:49

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