Let C be a hyperelliptic curve of genus $g \geq 3$ . In this paper, we give a new geometric description of the theta map for moduli spaces of rank 2 semistable vector bundles on C with trivial determinant. In order to do this, we describe a fibration of (a birational model of) the moduli space, whose fibers are GIT quotients $(\mathbb {P}^1)^{2g}//\text {PGL(2)}$ . Then, we identify the restriction of the theta map to these GIT quotients with some explicit degree 2 osculating projection. As a corollary of this construction, we obtain a birational inclusion of a fibration in Kummer $(g-1)$ -varieties over $\mathbb {P}^g$ inside the ramification locus of the theta map.

令C为属 $g \geq 3$ 的超椭圆曲线。在本文中，我们给出了C上具有平凡行列式的 2 阶半稳态向量丛模空间的 theta 映射的新几何描述。为了做到这一点，我们描述了模空间（的双有理模型）的纤维化，其纤维是 GIT 商 $(\mathbb {P}^1)^{2g}//\text {PGL(2)} $ . 然后，我们通过一些明确的 2 度密切投影确定了 theta 映射对这些 GIT 商的限制。作为这种构造的推论，我们在 theta 映射的分支轨迹内获得了 Kummer $(g-1)$ -varieties中的纤维化的双理性包含 $\mathbb {P}^g$ 。