We produce Brill-Noether general graphs in every genus, confirming a conjecture of Baker and giving a new proof of the Brill-Noether Theorem, due to Griffiths and Harris, over any algebraically closed field.

Abstract. Let C be an algebraic curve defined by a sufficiently generic bivariate Laurent polynomial with given Newton polygon ∆. It is classical that the geometric genus of C equals the number of lattice points in the interior of ∆. In this paper we give similar combinatorial interpretations for the gonality, the Clifford index and the Clifford dimension, by removing a technical assumption from a recent result of Kawaguchi. More generally, the method shows that apart from certain well-understood exceptions, every base-point free pencil whose degree equals or slightly exceeds the gonality is combinatorial, in the sense that it corresponds to projecting C along a lattice direction. Along the way we prove various features of combinatorial pencils. For instance, we give an interpretation for the scrollar invariants associated to a combinatorial pencil, and show how one can tell whether the pencil is complete or not.Among the applications, we find that every smooth projective curve admits at most one Weierstrass semi-group of embedding dimension 2, and that if a non-hyperelliptic smooth projective curve C of genus g ≥ 2 can be embedded in the n th Hirzebruch surface H n , then n is actually an invariant of C.

Let T be a weighted tree with n numbered leaves and let D = (D (i, j))i, j be its distance matrix, so D (i, j) is the distance between the leaves i and j. If m is an integer satisfying 2 ≤ m ≤ n, we prove a tropical formula to compute the m-dissimilarity map of T (i.e. the weights of the subtrees of T with m leaves), given D. For m = 3, we present a tropical description of the set of m-dissimilarity maps of trees. For m = 4, a partial result is given

We prove that in the moduli space of genus-g metric graphs the locus of graphs with gonality at most d has the classical dimension min{3g − 3, 2g + 2d − 5}. This follows from a careful parameter count to establish the upper bound and a construction of sufficiently many graphs with gonality at most d to establish the lower bound. Here, gonality is the minimal degree of a non-degenerate harmonic map to a tree that satisfies the Riemann-Hurwitz condition everywhere. Along the way, we establish a convenient combinatorial datum capturing such harmonic maps to trees.

In this paper we study plane curves of degree d with e total inflection points, for nonzero natural numbers d and e.Mathematics Subject Classification (2000). 14H10, 14H50, 14N20.Keywords. Plane curve, total inflection point, nodal curve. Introduction.Let e and d be nonzero natural numbers and denote by V d,e the set of elements ((L 1 , P 1 ), . . . , (L e , P e )) with L 1 , . . . , L e lines in the complex projective plane P 2 , P i a point on the line L i for each i, P i ∈ L j for each i = j and such that there exists a plane curve C of degree d with contact order d with L i at P i for each i. So P 1 , . . . , P e are total inflection points of C.In [5], the case d = 4 (i.e. quartic curves) has been studied intensively. The main tool used in that thesis is the so-called λ-invariant, which is nothing else than a cross ratio of four points. In [3], the cases e = 1, 2 have been handled and also the cases e = 3, 4 for some special configurations of the lines L i and the points P i .In Section 1 of this paper, we will first prove a general result (Proposition 1.3). Note that part (a) of this Proposition is already known (see [5, Chapter II, Lemma 2.15] or [3, Theorem A]). In Section 2, we prove a generalization of the main result of a paper of E. Ballico (see [2]). In Section 3, we introduce the notion of expected dimension for a component of V d,e .In a following paper of the authors, the established techniques of this paper will be used to describe all the components of V d,e in case e is equal to 3 or 4, together with their images in the moduli spaces. Both authors are partially supported by the Fund of Scientific Research -Flanders (G.0318.06). Definition and a general result.Definition 1.1. Let P 2 be the complex projective plane and P 2 be the incidence relation in (P 2 ) * × P 2 , thus P 2 = {(L, P )|P ∈ L}.