| This article analyzes the structure of Borel orbits in the subvariety of gln defined by X 2 = 0. The number of Borel orbits is finite, and they are in one to one correspondence with certain partial permutation matrices. Equations are found up to radical for the Zariski closure of each orbit and these equations are shown to be generically reduced. The orbits are given a poset structure, which can also be described in terms of certain words. The Zariski closure of an orbit can be determined from the poset. The dimension of an orbit (as an algebraic variety) is given by a rank function for the poset, which is defined in terms of a statistic of the word of an orbit. An algorithm for calculating the multidegree and group of Weil divisors of the Zariski closure of a given orbit is discussed. Restricting to the set of all orbits of a given Jordan type, we examine the action of the symmetric group on the set of representatives, which gives another explanation of the Borel orbit poset and allows us to develop Bott-Samuelson style desingularizations of the closures of Borel orbits. |
