On the connectedness of the scheme of multisecants to a projective curve
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)].
scientific article; zbMATH DE number 727315
scientific article