Projective toric varieties of codimension 2 with maximal Castelnuovo--Mumford regularity

The Eisenbud--Goto conjecture states that $\operatorname{reg} X\le\operatorname{deg} X -\operatorname{codim} X+1$ for a nondegenerate irreducible projective variety $X$ over an algebraically closed field. While this conjecture is known to be false in general, it has been proven in several special cases, including when $X$ is a projective toric variety of codimension $2$. We classify the projective toric varieties of codimension $2$ having maximal regularity, that is, for which equality holds in the Eisenbud--Goto bound. We also give combinatorial characterizations of the arithmetically Cohen--Macaulay toric varieties of maximal regularity in characteristic $0$.