Abstract
We study moduli spaces of rational weighted stable tropical curves, and their connections with Hassett spaces. Given a vector $w$ of weights, the moduli space of tropical $w$-stable curves can be given the structure of a balanced fan if and only if $w$ has only heavy and light entries. In this case, the tropical moduli space can be expressed as the Bergman fan of an explicit graphic matroid. The tropical moduli space can be realized as a geometric tropicalization, and as a Berkovich skeleton, its algebraic counterpart. This builds on previous work of Tevelev, Gibney and Maclagan, and Abramovich, Caporaso and Payne. Finally, we construct the moduli spaces of heavy/light weighted tropical curves as fibre products of unweighted spaces, and explore parallels with the algebraic world.
Highlights
We study tropical analogues of moduli spaces of rational weighted stable curves and their relation to the algebraic moduli spaces
We introduce tropical rational weighted stable curves in the natural way, by defining the combinatorial type of a w-stable tropical curve to be the dual graph of a w-stable curve, keeping track of the weights on the marked ends (Definition 2.1)
Gt (w) = Kn, the complete graph on n vertices, and the reduced graph Bergman fan corresponding to Theorem 2.17 tells us that the is G(w) = K this graph is combinatorial nsB−u1b(
Summary
The cone complex M0tr,owp can be given the structure of a balanced fan in a vector space if and only if w has only heavy and light entries. Since no ray of M0tr,onp becomes unstable, prw is the identity map In this case, Gt (w) = Kn, the complete graph on n vertices, and the reduced graph Bergman fan corresponding to Theorem 2.17 tells us that the is G(w) = K this graph is combinatorial nsB−u1b(. The following definition characterizes those cones that obstruct a balanced embedding of M0tr,owp the top-dimensional in a vector space Such cones will correspond precisely to cones of M0tr,onp on which prw is not injective; see Lemma 2.23. Suppose w is a weight vector with only heavy and light entries It follows ftfdhareocedmtuttochLpea-etfdmtrihommemesanmP2srai.oo2llpn3poatolshiicatniottosnpnearwsr2e.wi2lsi5eginhtwhjteaagctnuttUiavtrwoeano=ctnoeenUastlwr0lta.hccaIotntntMoeost0pthr,owwaepsrhissiwcfphoruorardmerse,dMipnmro0trw,toenpnicnstooinUotnMwraa.0ltcrW,.owtTps.eooTcsnhaelnyee this, consider a cone whose top-dimensional faces are contracted.
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