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We are working on a problem about semistable pure dimension one sheaves of rank $1$ and degree $0$ on a singular curve $C$ (for example, the Kodaira fiber of type $I_2$, i.e. $C=C_1\cup C_2$ where $C_i$ are rational smooth curves isomorphic to $\mathbb{P}^1$ with $C_1\cdot C_2=P+Q$), and this means that the sheaf is a line bundle away from the singular point, so we call it the semistable generalized line bundle of degree $0$.

From Ana Cristina Lopez's paper https://arxiv.org/abs/math/0410394, she described the moduli space of semistable generalized line bundle with more details in https://arxiv.org/abs/math/0410393. In particular, by Lemma 3.1 in the first paper, we know the graded object of strictly semistable generalized line bundle of degree $0$ is $Gr(F)=\mathcal{O}_{\mathbb{P}_1}(-1)\oplus\mathcal{O}_{\mathbb{P}_1}(-1)$, i.e. we shrink all the strictly semistable cases to one point by $S$-equivalence.

However for the future working, the most efficient way is to write out all the strictly semistable cases explicitly, so I am wondering if there is a way to do this.

Thanks for any help!

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  • $\begingroup$ Do you mean you want to describe semistable sheaves $F$ such that $Gr(F) = \mathcal{O}_{C_1}(-1) \oplus \mathcal{O}_{C_2}(-1)$? $\endgroup$
    – Sasha
    Commented Oct 25, 2022 at 5:02
  • $\begingroup$ @Sasha Thanks for your comment. In this case we require the degree of $F$ to be $0$, so it is a subset of the sheaves that you described. $\endgroup$
    – Ruoxi Li
    Commented Oct 25, 2022 at 5:26
  • $\begingroup$ Your answer seems to be negative --- then I don't understand your question. Can you formulate it more precisely? $\endgroup$
    – Sasha
    Commented Oct 25, 2022 at 6:22
  • $\begingroup$ @Sasha We want to describe all the strictly semistable sheaves on $C$, in this case we have $F\vert_{C_1}\cong\mathcal{O}_{\mathbb{P}_1}(-1)$ and $F\vert_{C_2}\cong\mathcal{O}_{\mathbb{P}_1}(1)$. Is it possible to write them explicitly? $\endgroup$
    – Ruoxi Li
    Commented Oct 25, 2022 at 7:19
  • $\begingroup$ Do you mean $F\vert_{C_2} \cong \mathcal{O}_{\mathbb{P}^1}(-1)$? $\endgroup$
    – Sasha
    Commented Oct 25, 2022 at 8:21

1 Answer 1

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Let $C_1 \cap C_2 = \{P,Q\}$. For any line bundles $L_1$ on $C_1$ and $L_2$ on $C_2$ any sheaf $F$ on $C$ such that $F\vert_{C_i} \cong L_i$ fits into an exact sequence $$ 0 \to F \to L_1 \oplus L_2 \to \mathcal{O}_P \oplus \mathcal{O}_Q \to 0, $$ where the morphisms $L_i \to \mathcal{O}_P \oplus \mathcal{O}_Q$ are surjective. In particular, it is invertible, and thus, such sheaves $F$ are parameterized by a principal homogeneous space over the group $$ \mathrm{Pic}^0(C) \cong \mathbb{G}_{\mathrm{m}}. $$

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