On the tangent bundle and the divisor theory of a general matroid

Abstract Extending classical algebro-geometric constructions to arbitrary matroids, we construct a $K$-class $T_{M}\in K(M)$ for every loopless matroid $M$. When $M$ is realizable by a linear subspace $L$, $T_{M}$ recovers the $K$-class of the tangent bundle of the wonderful compactification $W_{L}$. We derive two formulas for the total Chern class of $T_{M}$ (one combinatorial and one geometric) and show that the associated Todd class agrees with the Todd class appearing in the matroid Hirzebruch–Riemann–Roch formula. To develop a positivity theory entirely at the combinatorial level, we introduce the notion of a “fake effective cone,” a combinatorial analogue of the classical effective cone, and use it to characterize big and nef divisors in $A(M)$. Finally, we define the $\beta _{S}$ classes, obtained from Cremona conjugates of the classical $\alpha _{S}$ classes, and study their properties to provide a rich and computable family of combinatorially nef divisors.