Let Fλa be the PBW degeneration of the flag varieties of type An-1. These varieties are singular and are acted upon with the degenerate Lie group SLna. We prove that Fλa have rational singularities, are normal and locally complete intersections, and construct a desingularization Rλ of Fλa. The varieties Rλ can be viewed as towers of successive ℙ1-fibrations, thus providing an analogue of the classical Bott-Samelson-Demazure-Hansen desingularization. We prove that the varieties Rλ are Frobenius split. This gives us Frobenius splitting for the degenerate flag varieties and allows to prove the Borel-Weil type theorem for Fλa. Using the Atiyah-Bott-Lefschetz formula for Rλ, we compute the q-characters of the highest weight sln-modules.