AI 中文总结
研究偶自旋曲线模空间中斯科尔扎对应关系的推广，先给出经典斯科尔扎曲线光滑性新证，再引入高阶类似物研究其几何性质，最后考虑三维高维版本，描述其类并证光滑性。
AI 中文摘要
我们研究与偶自旋曲线模空间$S_g^+$中一般的$(C,\eta)$相关的斯科尔扎对应关系的推广。首先给出由法卡斯 - 韦拉最初建立的经典斯科尔扎曲线光滑性的新证明。接着引入斯科尔扎对应关系的高阶类似物并研究其几何性质，包括计算其在 Néron - Severi 群中的类、研究奇点和亏格。最后考虑该构造的高维版本，聚焦三维情形，描述其类并证明在自然定义的退化轨迹之外的光滑性结果。
英文摘要
We study generalizations of the Scorza correspondence associated with a general $(C,η)$ in the moduli space of even spin curves $S_g^+$. We first give a new proof of the smoothness of the classical Scorza curve, originally established by Farkas-Verra. We then introduce higher order analogues of the Scorza correspondence and investigate their geometry. In particular, we compute their classes in the Néron-Severi group and study their singularities and genera. Finally, we consider a higher-dimensional version of the construction and focus on the threefold case, where we describe its class and prove a smoothness result outside a naturally defined degeneracy locus.
