Boris Lerner

About me

I am currently a Japanese Society for the Promotion of Science (JSPS) postdoctoral fellow at Nagoya University working with Professor Osamu Iyama . I completed my PhD at the University of New South Wales (UNSW) in 2012 under the close supervision and guidance of Dr. Daniel Chan. My field of research is noncommutative algebraic geometry.

Research

My main interest is orders on surfaces, which are simply coherent torsion free sheaves of algebras (on surfaces) that are generically central simple. These objects play an analogous role, in the study of various noncommutative rings, to that of varieties in the study of function fields, i.e. they form integral models. The beautiful feature of orders is that the geometry of the surface has a large effect on their properties. Thus the field utilises a great deal of classical algebraic geometry in this noncommutative setting. Here are two great places to learn about orders: Lectures on Orders  by Daniel Chan and Stable Orders on Surfaces by Artin and de Jong.
This is my PhD thesis: Line Bundles and Curves on a del Pezzo Order (thesis)

Papers

1. Line Bundles and Curves on a del Pezzo Order. In Journal of Algebra, Volume 387, 1 August 2013, Pages 117–143.

   Abstract:
   Orders on surfaces provide a rich source of examples of noncommutative surfaces. In [1] the authors prove the existence of the analogue of the Picard scheme for orders and in [2] the Picard scheme is explicitly computed for an order on the projective plane ramified on a smooth quartic. In this paper, we continue this line of work, by studying the Picard and Hilbert schemes for an order on the projective plane ramified on a union of two conics. Our main result is that, upon carefully selecting the right Chern classes, the Hilbert scheme is a ruled surface over a genus two curve. Furthermore, this genus two curve is, in itself, the Picard scheme of the order.  

2. Tilting Bundles on Orders on Projective Space. (To appear in the Israel Journal of Mathematics).

   Abstract:
   We introduce a class of orders on $\P^d$ called Geigle-Lenzing orders and show that they have tilting bundles. Moreover we show that their module categories are equivalent to the categories of coherent sheaves on Geigle-Lenzing spaces introduced by Herschend, Iyama, Minamoto and Oppermann.

Recent talks

1. Nagoya-Warwick workshop on McKay correspondence. Warwick, February 2014.