Syzygies of tangent-developable surfaces and K3 carpets via secant varieties

Park, Jinhyung

doi:10.2140/ant.2025.19.1029

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Syzygies of tangent-developable surfaces and K3 carpets via secant varieties

Jinhyung Park

Vol. 19 (2025), No. 5, 1029–1048

DOI: 10.2140/ant.2025.19.1029

Abstract

We give simple geometric proofs of Aprodu, Farkas, Papadima, Raicu and Weyman’s theorem on syzygies of tangent-developable surfaces of rational normal curves and Raicu and Sam’s result on syzygies of K3 carpets. As a consequence, we obtain a quick proof of Green’s conjecture for general curves of genus $g$ over an algebraically closed field $k$ with $char (k) = 0$ or $char (k) \geq ⌊ (g - 1) ∕ 2 ⌋$ . Our approach provides a new way to study tangent-developable surfaces in general. Along the way, we show the arithmetic normality of tangent-developable surfaces of arbitrary smooth projective curves of large degree.

Keywords

tangent-developable surfaces, K3 carpets, secant varieties, syzygies

Mathematical Subject Classification

Primary: 14N05, 14N07, 13D02

Milestones

Received: 28 April 2024

Accepted: 16 July 2024

Published: 22 April 2025

Authors

Jinhyung Park
Department of Mathematical Sciences KAIST Daejeon South Korea

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Vol. 19, No. 5, 2025