A b s t r ac t . We study the fan structure of Dressians Dr (d, n) and local Dressians Dr(M) for a given matroid M. In particular we show that the fan structure on Dr(M) given by the three term Plücker relations coincides with the structure as a subfan of the secondary fan of the matroid polytope P (M). As a corollary, we have that a matroid subdivision is determined by its 3-dimensional skeleton. We also prove that the Dressian of the sum of two matroids is isomorphic to the product of the Dressians of the matroids. Finally we focus on indecomposable matroids. We show that binary matroids are indecomposable, and we provide a non-binary indecomposable matroid as a counterexample for the converse. arXiv:1809.08965v1 [math.CO] 24 Sep 2018 Cambridge University Press, Cambridge, 1986.I n s t i t u t f ü r M at h e m at i k , F U B e r l i n , A r n i m a l l e e 2 , 1 4 1 9 5 B e r l i n , G e r m a n y , E -m a i l : o l a rt e @ z e dat . f u -b e r l i n . d e I n s t i t u t f ü r M at h e m at i k , T U B e r l i n , S t r . d e s 1 7 . J u n i 1 3 6 , 1 0 6 2 3 B e r l i n , G e r m a n y , E -m a i l : pa n i z z u t @ m at h . t u -b e r l i n . d e D e pa rt m e n t o f M at h e m at i c a l S c i e n c e s , B i n g h a m t o n U n i v e r s i t y , B i n g h a m t o n , N Y 1 3 9 0 2 , U S A , E -m a i l : s c h ro e t e r @ m at h . b i n g h a m t o n . e d u

Abstract. We study theta characteristics of hyperelliptic metric graphs of genus g with no bridge edges. These graphs have a harmonic morphism of degree two to a metric tree that can be lifted to morphism of degree two of a hyperelliptic curve X over K to the projective line, with K an algebraically closed field of char(K) = 2, complete with respect to a non-Archimedean valuation, with residue field k of char(k) = 2. The hyperelliptic curve has 2 2g theta characteristics. We show that for each effective theta characteristics on the graph, 2 g−1 even and 2 g−1 odd theta characteristics on the curve specialize to it; and 2 g even theta characteristics on the curve specialize to the unique not effective theta characteristics on the graph.

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The gonality sequence $(\gamma_r)_{r\geq1}$ of a finite graph/metric graph/algebraic curve comprises the minimal degrees $\gamma_r$ of linear systems of rank $r$. For the complete graph $K_d$, we show that $\gamma_r =  kd - h$ if $r<g=\frac{(d-1)(d-2)}{2}$, where $k$ and $h$ are the uniquely determined integers such that $r = \frac{k(k+3)}{2} - h$ with $1\leq k\leq d-3$ and $0 \leq h \leq k $. This shows that the graph $K_d$ has the gonality sequence of a smooth plane curve of degree $d$. The same result holds for the corresponding metric graphs.

Smooth tropical cubic surfaces are parametrized by maximal cones in the unimodular secondary fan of the triple tetrahedron. There are $$344\, 843 \,867$$ 344 843 867 such cones, organized into a database of $$14\,373\,645$$ 14 373 645 symmetry classes. The Schläfli fan gives a further refinement of these cones. It reveals all possible patterns of lines on tropical cubic surfaces, thus serving as a combinatorial base space for the universal Fano variety. This article develops the relevant theory and offers a blueprint for the analysis of big data in tropical geometry.