Hilbert scheme of linearly normal curves in ℙ^r with index of speciality five and beyond

2023/12/03 by Changho Keem, Keem, Changho
Arts and Humanities · Mathematics · #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Meromorphic and Entire Functions #North African History and Literature #Primary 14C05 #Secondary 14H10

paper · pdf · doi:10.48550/arxiv.2312.01347

openalex publication_date 2023/12/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study the Hilbert scheme of smooth, irreducible, non-degenerate and linearly normal curves of degree d and genus g in ℙ^r (r≥ 3) whose complete and very ample hyperplane linear series D have relatively small index of speciality i(D)=g-d+r. In particular we determine the existence as well as the non-existence of Hilbert schemes of linearly normal curves H^L_d,g,r for every possible triples (d,g,r) with i(D)=5 and r≥ 3. We also determine the irreducibility of the Hilbert scheme H^L_g+r-5,g,r when the genus g is near to the minimal possible value with respect to the dimension of the projective space ℙ^r for which H^L_g+r-5,g,r≠∅, say r+9≤ g≤ r+11. In the course of proofs of key results, we show the existence of linearly normal curves of degree d≥ g+1 with arbitrarily given index of speciality with some mild restriction on the genus g.

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