Han-Bom Moon

Birational Geometry of the Moduli Space of Rank 2 Parabolic Vector Bundles on a Rational Curve

ABSTRACT. We investigate the birational geometry (in the sense of Mori’s program) of the moduli space of rank 2 semistable parabolic vector bundles on a rational curve. We compute the effective cone of the moduli space and show that all birational models obtained by Mori’s program are also moduli spaces of parabolic vector bundles with certain parabolic weights. In this talk, we introduce wall-crossings of the moduli space, sl2-conformal blocks and double sequences that are central techniques for the computation of the effective cone. This is a joint work with Dr. Han-Bom Moon.

Birational contractions of \overline{\mathrm {M}}_{0,n} and combinatorics of extremal assignments

From Smyth’s classification, modular compactifications of the moduli space of pointed smooth rational curves are indexed by combinatorial data, the so-called extremal assignments. We explore their combinatorial structures and show that any extremal assignment is a finite union of atomic extremal assignments. We discuss a connection with the birational geometry of the moduli space of stable pointed rational curves. As applications, we study three special classes of extremal assignments: smooth, toric, and invariant with respect to the symmetric group action. We identify them with three combinatorial objects: simple intersecting families, complete multipartite graphs, and special families of integer partitions, respectively.