AI 中文总结
研究光滑射影曲线高割线簇的行列式表示，利用高阶Szegő核等工具，在嵌入充裕时给出对称表示，对实曲线割线簇给出正定条件，对有理正规曲线割线簇给出明确表示，还能在特定条件下将曲线凸包表示为谱面体。
AI 中文摘要
我们证明，如果嵌入足够充裕，光滑射影曲线的高割线簇具有对称可允许行列式表示，其秩为一的对称Ulrich层。对于实曲线的割线簇，我们给出表示矩阵正定的条件。在温和（推测为空）条件下，当曲线的凸包为双曲性锥时，我们可将其表示为谱面体。对于有理正规曲线的割线簇，我们根据Littlewood - Richardson系数得到非常明确的表示。我们使用的一个关键工具是与曲线上非有效theta特征相关的高阶Szegő核和高阶Scorza对应。
英文摘要
We show that higher secant varieties of smooth projective curves have symmetric admissible determinantal representations with symmetric Ulrich sheaves of rank one if the embedding is sufficiently ample. For secant varieties of real curves, we give conditions for the representing matrices to be positive definite. This allows us, under mild (conjecturally vacuous) conditions, to represent the convex hull of the curve as a spectrahedron whenever it is a hyperbolicity cone. For secant varieties of rational normal curves, we derive very explicit representations in terms of Littlewood--Richardson coefficients. One key tool that we use are the higher Szegő kernels and the higher Scorza correspondences associated to a non-effective theta characteristic on the curve.
Comments47 pages, comments very welcome
