Singularities and syzygies of secant varieties of nonsingular projective curves
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Singularities and syzygies of secant varieties of nonsingular projective curves Lawrence Ein1 · Wenbo Niu2 · Jinhyung Park3
Received: 2 May 2019 / Accepted: 14 May 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract 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 genus g is greater than 2g + 2k + p for nonnegative integers k and p, then the k-th secant variety of the curve has normal Du Bois singularities, is arithmetically Cohen–Macaulay, and satisfies the property Nk+2, p . In addition, the singularities of the secant varieties are further classified according to the genus of the curve, and the Castelnuovo–Mumford
L. Ein was partially supported by NSF Grant DMS-1801870. J. Park was partially supported by NRF-2016R1C1B2011446 and the Sogang University Research Grant of 201910002.01.
B Lawrence Ein [email protected]
Wenbo Niu [email protected] Jinhyung Park [email protected] 1
Department of Mathematics, University Illinois at Chicago, 851 South Morgan St., Chicago, IL 60607, USA
2
Department of Mathematical Sciences, University of Arkansas, Fayetteville, AR 72701, USA
3
Department of Mathematics, Sogang University, 35 Beakbeom-ro, Mapo-gu, Seoul 04107, Republic of Korea
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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. Mathematics Subject Classification 13A10 · 14Q20 Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Singularities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Syzygies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Symmetric products, secant bundles, and secant varieties . . . . . . . . . . . 3.1 Lemmas on symmetric products . . . . . . . . . . . . . . . . . . . . . 3.2 Secant varieties via secant bundles . . . . . . . . . . . . . . . . . . . . 3.3 Blowup construction of secant bundles . . . . . . . . . . . . . . . . . . 4 A vanishing theorem on Cartesian products of curves . . . . . . . . . . . . . 5 Properties of secant varieties of curves . . . . . . . . . . . . . . . . . . . . 5.1 Normality, projective normality, and property Nk+2, p . . . . . . . . . . 5.2 Singularities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Arithmetic Cohen–Macaulayness and Castelnuovo–Mumford regularity 5.4 Further properties of secant varieties . . . . . . . . . . . . . . . . . . . 6 Open problems . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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