Before revision:  As an application, we revisit the problem of eliminating depth cycles among non-vertical lines in 3-space, recently studied by Aronov and Sharir (2018), and show an algorithm that cuts n such lines into O(n^{3/2+ \varepsilon }) pieces that are depth-cycle free, for any \varepsilon > 0.
After revision:  As an application, we revisit the problem of eliminating depth cycles among non-vertical lines in 3-space, recently studied by Aronov and Sharir (2018), and show an algorithm that cuts n such lines into O(n^{3/2+ \epsilon }) pieces that are depth-cycle free, for any \varepsilon > 0.