# A brill - noether theory for k-gonal nodal curves by Ballico E.

By Ballico E.

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1), is Diff(S 1 )-invariant. Proof. 7). We need to check that the map A → L commutes with the Diff(S 1 )-action. 2. 3) is a linear differential operator with the (n − 1)-order 44 CHAPTER 2. 1). We have already encountered a similar formula: this is the Diff(S 1 )action on the coefficient of a Sturm-Liouville operator L = c(d/dx) 2 + an−1 , cf. 7) in which, however, c = 1. 1) defines a projective structure on γ(x). 1 implies that this projective structure does not depend on the parameterization.

If one changes the parameter by a fractional-linear transformation x → (ax + b)/(cx + d), the resulting curve is projectively equivalent to the original one. In appropriate affine coordinates, this curve is given by γ = (1 : x : x2 : · · · : xk−1 ). 5) Dual operators and dual curves Given a linear differential operator A : F λ → Fµ on S 1 , its dual operator A∗ : F1−µ → F1−λ is defined by the equality φA∗ (ψ) A(φ)ψ = S1 S1 for any φ ∈ Fλ and ψ ∈ F1−µ . The operation A → A∗ is Diff(S 1 )-invariant.

7. Prove the following explicit formula: B(φ, ψ) = (−1)r+t+1 r+s+t≤n r + t (r) ar+s+t+1 φ(s) ψ (t) . 8. , a symplectic structure, on the space Ker A – cf. [166]. Monodromy If γ is a closed curve, then the operator A has periodic coefficients. The converse is not at all true. Let A be an operator with periodic coefficients, in other words, a differential operator on S 1 . The solutions of the equation Aφ = 0 are not necessarily periodic; they are defined on R, viewed as the universal covering of S 1 = R/2πZ.