Determinantal equations for curves of high degree. (English) Zbl 0681.14027

Eisenbud, David; Koh, Jee; Stillman, Michael

Am. J. Math. 110, No. 3, 513-539 (1988).

Let C be a reduced, irreducible curve of arithmetic genus g and let \(L_ 1, L_ 2\) be two line bundles on C of degree \(\geq 2g+1,\) nonisomorphic if \(g>0\) and both have degree \(2g+1.\) If \({\mathfrak L}_ i\), \(i=1,2\), denotes the complete linear series \((L_ i,H^ 0(L_ i))\) on C, \({\mathfrak L}_ 1{\mathfrak L}_ 2\) the product series \((L_ 1\otimes L_ 2,V)\) and \(\phi_{{\mathfrak L}_ 1{\mathfrak L}_ 2}\) the associated rational map from C to \({\mathbb{P}}(V)\), then the main result (see theorem 1) of this paper says that the (maximal) homogeneous ideal of \(\phi_{{\mathfrak L}_ 1{\mathfrak L}_ 2}(C)\) is generated by the \(2\times 2\)-minors of a matrix of linear forms. This is an extension of a corresponding result of Castelnuovo (see theorem).
As a corollary one gets that any curve of genus \(g>0,\) embedded by a complete linear series of degree \(4g+2\) into \({\mathbb{P}}^{3g+2}\) has equations which may be realized as the \(2\times 2\) minors of a \((g+2)\times (g+2)\) matrix of linear forms. Examples of the determinantal representations of elliptic and hyperelliptic curves are considered. In an appendix a generalized form of Clifford’s theorem (for singular curves) is given (see theorem A).

Reviewer: M.Herrmann

MSC:

14M12	Determinantal varieties
14H45	Special algebraic curves and curves of low genus
13A15	Ideals and multiplicative ideal theory in commutative rings

Keywords:

determinantal equations for curves; minors of a matrix of linear forms