The goal of this paper is the study of simple rank 2 parabolic vector bundles over a $2$-punctured elliptic curve $C$. We show that the moduli space of these bundles is a non-separated gluing of two charts isomorphic to $\mathbb{P}^1 \times \mathbb{P}^1$. We also showcase a special curve $\Gamma$ isomorphic to $C$ embedded in this space, and this way we prove a Torelli theorem. This moduli space is related to the moduli space of semistable parabolic bundles over $\mathbb{P}^1$ via a modular map which turns out to be the 2:1 cover ramified in $\Gamma$. We recover the geometry of del Pezzo surfaces of degree 4 and we reconstruct all their automorphisms via elementary transformations of parabolic vector bundles.

中文翻译：

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椭圆曲线上抛物线丛模的几何

本文的目标是研究 $2$ 穿孔椭圆曲线 $C$ 上的简单 2 阶抛物线向量丛。我们证明这些包的模空间是同构于 $\mathbb{P}^1 \times \mathbb{P}^1$ 的两个图表的非分离粘合。我们还展示了嵌入该空间的与 $C$ 同构的特殊曲线 $\Gamma$，这样我们证明了 Torelli 定理。这个模空间通过模映射与 $\mathbb{P}^1$ 上的半稳定抛物线丛的模空间相关，模映射结果是 $\Gamma$ 中的 2:1 覆盖。我们恢复了 4 次 del Pezzo 曲面的几何形状，并通过抛物线向量丛的初等变换重建了它们的所有自同构。