Singularities and syzygies of secant varieties of nonsingular projective curves

• 成果类型: Article
• 署名作者: Ein, Lawrence; Niu, Wenbo; Park, Jinhyung
• 署名单位: University of Illinois System; University of Illinois Chicago; University of Illinois Chicago Hospital; University of Arkansas System; University of Arkansas Fayetteville; Sogang University
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-020-00976-5
• 发表日期: 2020
• 页码: 615-665
• 关键词: koszul cohomology; geometry; REGULARITY; normality; EQUATIONS

摘要

In recent years, the equations defining secant varieties and their syzygies have attracted considerable attention. The purpose of the present paper is to conduct a thorough study on secant varieties of curves by settling several conjectures and revealing interaction between singularities and syzygies. The main results assert that if the degree of the embedding line bundle of a nonsingular curve of genusgis greater than 2g+2k+p for nonnegative integerskandp, then thek-th secant variety of the curve has normal Du Bois singularities, is arithmetically Cohen-Macaulay, and satisfies the property N-k+2,N-p. In addition, the singularities of the secant varieties are further classified according to the genus of the curve, and the Castelnuovo-Mumford regularities are also obtained as well. As one of the main technical ingredients, we establish a vanishing theorem on the Cartesian products of the curve, which may have independent interests and may find applications elsewhere.