We compute toric degenerations arising from the tropicalization of the full flag varieties Fℓ 4 and Fℓ 5 embedded in a product of Grassmannians. For Fℓ 4 and Fℓ 5 we compare toric degenerations arising from string polytopes and the FFLV polytope with those obtained from the tropicalization of the flag varieties. We also present a general procedure to find toric degenerations in the cases where the initial ideal arising from a cone of the tropicalization of a variety is not prime.

We define a tropical version Fd(tropX)$\operatorname{F}_d(\operatorname{trop}X)$ of the Fano Scheme Fd(X)$\operatorname{F}_d(X)$ of a projective variety X⊆Pn$X\subseteq \mathbb {P}^n$ and prove that Fd(tropX)$\operatorname{F}_d(\operatorname{trop}X)$ is the support of a polyhedral complex contained in prefixtropdouble-struckGfalse(d,nfalse)$\operatorname{trop}\mathbb {G}(d,n)$. In general, prefixtropFd(X)⊆Fd(tropX)$\operatorname{trop}\operatorname{F}_d(X)\subseteq \operatorname{F}_d(\operatorname{trop}X)$ but we construct linear spaces L$L$ such that prefixtropF1(X)⊊F1(tropX)$\operatorname{trop}\operatorname{F}_1(X)\subsetneq \operatorname{F}_1(\operatorname{trop}X)$ and show that for a toric variety prefixtropFd(X)=Fd(tropX)$\operatorname{trop}\operatorname{F}_d(X)=\operatorname{F}_d(\operatorname{trop}X)$.

In this paper we study the interplay between tropical convexity and its classical counterpart. In particular, we focus on the tropical convex hull of convex sets and polyhedral complexes. We give a vertex description of the tropical convex hull of a line segment and of a ray in R n /R1 and show that tropical convex hull and classical convex hull commute in R 3 /R1. Finally we prove results on the dimension of tropically convex fans and give a lower bound on the degree of a tropical curve under certain hypotheses.

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