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