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

Abstract: For a loopless matroid $M$, we construct a $K$-class $T_M\in K(X_{M})$. When $M$ is realizable, $T_M$ recovers the $K$-class of the tangent bundle of the wonderful compactification $W_L$. We derive two formulas for the total Chern class of $T_M$ and prove that the associated Todd class agrees with the Todd class appearing in the matroid Hirzebruch--Riemann--Roch formula. We define a ``fake effective cone'' so that big and nef divisors in a matroid can be characterized in a manner analogous to how the effective cone characterizes big and nef divisors in classical algebraic geometry. Finally, we define the classes $\beta_S$ and study their properties.