Irreducibility and components rigid in moduli of the Hilbert scheme of\n smooth curves

2016/05/01 by Changho Keem, Keem, Changho, Yun-Hwan Kim +3
Mathematics · #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometric Analysis and Curvature Flows #Meromorphic and Entire Functions

paper · pdf · doi:10.48550/arxiv.1605.00297

openalex publication_date 2016/05/01 · openalex created_date 2022/10/03 · openalex updated_date 2026/07/28

Abstract

Denote by \H_d,g,r the Hilbert scheme of smooth curves, that is\nthe union of components whose general point corresponds to a smooth irreducible\nand non-degenerate curve of degree d and genus g in mathbb P^r. A\ncomponent of \H_d,g,r is rigid in moduli if its image under the\nnatural map \π:\H_d,g,r dashrightarrow \M_g is a one\npoint set. In this note, we provide a proof of the fact that\n\H_d,g,r has no components rigid in moduli for g > 0 and r=3,\nfrom which it follows that the only smooth projective curves embedded in\n mathbb P³ whose only deformations are given by projective transformations\nare the twisted cubic curves. In case r \≥ 4, we also prove the\nnon-existence of a component of \H_d,g,r rigid in moduli in a\ncertain restricted range of d, g>0 and r. In the course of the proofs, we\nestablish the irreducibility of \H_d,g,3 beyond the range which\nhas been known before.\n

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