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Brill-Noether theory of binary curves. (English) Zbl 1237.14033

The fundamental theorems of classical Brill-Noether theory of smooth projective curves have not been extended yet to stable curves, due to the technical difficulties of combinatorial nature. The goal of this paper is to extend some of them to binary curves. A binary curve is a stable nodal curve having two irreducible rational components, intersecting at \(g+1\) points. Their moduli space \(B_g\subset \overline{M_g}\) is irreducible of dimension \(2g-4\). The analogue of theorems of Riemann, Clifford and Martens are proved to hold for any binary curve and for line bundles parametrized by the compactified Jacobian scheme. An analogue of Brill-Noether theorem is proved for general binary curves and for \(r\leq 2\). More precisely, let \[ B_d(g)=\{(d_1,d_2) \mid d_1+d_2=d, \frac{d-g-1}{2}\leq d_i\leq \frac{d-g+1}{2}, i=1,2\} \] be the set of balanced multidegrees.
If \(\underline{d}\in B_d(g)\), put \(W^r_{\underline{d}}(X)= \{L\in Pic^{\underline{d}}(X)\mid h^o(L)\geq r+1\}\), where \(X\) is a binary curve. Then it is proved that, for a general binary curve \(X\) and for \(r\leq 2\), \(\dim W^r_{\underline{d}}(X)\leq \rho_d^r(g)\), the usual Brill-Noether number, and equality holds for some \(\underline{d}\). Moreover \(\dim \overline{W^r_{\underline{d}}(X)}= \rho_d^r(g)\).

MSC:

14H10 Families, moduli of curves (algebraic)
14C20 Divisors, linear systems, invertible sheaves
14H51 Special divisors on curves (gonality, Brill-Noether theory)
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