On the connectedness of the scheme of multisecants to a projective curve
type: Item
instance of: scholarly article
title: On the connectedness of the scheme of multisecants to a projective curve
zbMATH Open document ID: 0833.14020
DOI: 10.1007/BF01264005
author: E. Ballico
published in: Geometriae Dedicata
publication date: 1995-02-26T00:00:00Z
review text: The set of multisecant spaces to a given projective variety is a classic subject of study (especially in the case of curves), and also in recent times it has been considered and worked upon in many papers for different reasons (enumerative geometry point of view, study of special divisors). This paper deals with the problem of determining the connectedness of this set, and the main result gives numerical conditions on \(a,e,t,n\) in order to have that the set of \(e\)-secant \(a\)-planes to a curve \(C\) in \(\mathbb{P}^n\) is connected when it is at most \(t\)-dimensional. From this the author also proves the following: Let \(C \subseteq \mathbb{P}^3_k\), \(k = \overline k\), be a smooth connected algebraic curve, with degree \(d\) and genus \(g\); let \(S\) be the scheme of its trisecant lines. Then in any of the following cases \(S\) is connected: (1) \(h^1 (C, {\mathcal O}_C(1)) = 0\); (2) \(\text{char} (k) = 0\), \(d \geq g + 4\), and \(g \geq 8\) if \(d = g + 4\); (3) \(\text{char} (k) = 0\), \(d = g + 3\), \(g \geq 14\) and \(C\) is a normal point of \(\text{Hilb}_{d,g} (\mathbb{P}^3)\). The main ingredients in the proof of these results are the Schwartzenberger's rk\(e\)-vector bundle on the \(e\)-th symmetric product of \(C\), the connectedness theorem of \textit{W. Fulton} and \textit{R. Lazarsfeld} [Acta Math. 146, 271-283 (1981; Zbl 0469.14018)], and some recent work by \textit{F. Laytimi} [Math. Ann. 294, No. 3, 459-462 (1992; Zbl 0757.14019)].
reviewed by: Alessandro Gimigliano
Mathematics Subject Classification ID: 14H50
Mathematics Subject Classification ID: 54D05
Mathematics Subject Classification ID: 14F45
Mathematics Subject Classification ID: 14N10
zbMATH DE Number: 727315
zbMATH Keywords: connectedness of multisecant spaces
MaRDI profile type: Publication
cites work: Q3347989
cites work: Q3812360
cites work: Q3977656
cites work: Hilbert scheme of smooth space curves
cites work: Divisors on general curves and cuspidal rational curves
cites work: On the connectedness of degeneracy loci and special divisors
cites work: Trisecant curves at a smooth curve in \(\mathbb{P}_ 3\) of degree \(d>2g+1\)
cites work: Secant Bundles on Symmetric Products
cites work: Vanishing theorems on complex manifolds
full work available at URL: https://doi.org/10.1007/bf01264005
OpenAlex ID: W1975990870
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description: scientific article; zbMATH DE number 727315