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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 ⁡ { 3 g − 3 , 2 g + 2 d − 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. [ABSTRACT FROM AUTHOR]

We give upper bounds on the minimal degree of a model in P 2 and the minimal bidegree of a model in P 1 × P 1 of the curve defined by a given Laurent polynomial, in terms of the combinatorics of the Newton polygon of the latter. We prove in various cases that this bound is sharp as soon as the polynomial is sufficiently generic with respect to its Newton polygon. [ABSTRACT FROM AUTHOR]

Abstract: 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. [Copyright &y& Elsevier]

Abstract: A point P on a smooth hypersurface X of degree d in is called a star point if and only if the intersection of X with the embedded tangent space is a cone with vertex P. This notion is a generalization of total inflection points on plane curves and Eckardt points on smooth cubic surfaces in . We generalize results on the configuration space of total inflection points on plane curves to star points. We give a detailed description of the configuration space for hypersurfaces with two or three star points. We investigate collinear star points and we prove that the number of star points on a smooth hypersurface is finite. [Copyright &y& Elsevier]

Abstract: In this article, we will prove that the set of 4-dissimilarity vectors of -trees is contained in the tropical Grassmannian . We will also propose three equivalent conjectures related to the set of -dissimilarity vectors of -trees for the case . Using a computer algebra system, we can prove these conjectures for . [Copyright &y& Elsevier]