On the existence of components of the Hilbert scheme with the expected number of moduli
type: Item
instance of: scholarly article
title: On the existence of components of the Hilbert scheme with the expected number of moduli
zbMATH Open document ID: 0702.14018
DOI: 10.1007/BF01446586
author: Angelo Felice Lopez
published in: Mathematische Annalen
publication date: 1991-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/164795
review text: Let W be an open irreducible subset of the Hilbert scheme of \({\mathbb{P}}^ r\) parametrizing smooth irreducible curves of degree n and genus g and \(\pi\) : \(W\to {\mathcal M}_ g\) the natural map to the moduli space of curves of genus g. W is said to have the expected number of moduli if \(\dim(\pi (W))=\min \{3g-3,3g-3+\rho \}\), where \(\rho =\rho (g,n,r)\) is the Brill-Noether number. The purpose of the paper is the construction of such families of curves for negative \(\rho\) 's, extending the previous existence range shown by \textit{E. Sernesi} [Invent. Math. 75, 25-57 (1984; Zbl 0541.14024)]. For \(r\geq 12\) or \(r=10\), it is shown the existence of families having the expected number of moduli in the range \(-(5r+\epsilon)g/(4r+9- \epsilon)+f(r)\leq \rho \leq 0\) where \(\epsilon\) is defined by \((r- \epsilon)/3=\lceil (r-1)/3\rceil\) and f(r) is a rational function of r asymptotically like \(5/4r^ 2\) (for the explicit expression see section 1 of the paper). - Note that the range where the existence problem makes sense is \(\rho \geq -3g+3\) and that the above result gives an affirmative answer roughly for \(\rho \geq -5(g-r^ 2)/4.\) The proof has two parts: (a) A careful study of the normal bundle of general nonspecial curves in \({\mathbb{P}}^ r\). This is accomplished by studying the special case of nonspecial curves lying in suitable rational normal surface scrolls; (b) an inductive proof starting from curves \(C\subset {\mathbb{P}}^ r\) whose existence was proved by E. Sernesi [loc. cit.] and attaching general nonspecial curves \(\Gamma\) at \(r+4\) general points. The results in (a) are then used to show that this construction is possible and that the curves \(C\cup \Gamma\) are flatly smoothable (the latter by standard deformation theory techniques developed by E. Sernesi [loc. cit.]).
Mathematics Subject Classification ID: 14H10
Mathematics Subject Classification ID: 14C05
zbMATH DE Number: 4150379
zbMATH Keywords: Hilbert scheme
zbMATH Keywords: expected number of moduli
zbMATH Keywords: Brill-Noether number
Wikidata QID: Q60143223
reviewed by: Antonio Fernández López
MaRDI profile type: Publication
cites work: Q3347989
cites work: Q3497199
cites work: Limit linear series: Basic theory
cites work: Q4143433
cites work: Components of the Hilbert scheme of smooth space curves with the expected number of moduli
cites work: Q3960945
cites work: On the existence of certain families of curves
OpenAlex ID: W2087711891
description: scientific article; zbMATH DE number 4150379