Jason Meintjes

San Francisco State University

“Projective Embeddings of the Moduli Space of Genus-Zero Curves”

The original construction of the compactified moduli space of n-pointed genus-zero curves due to Knudsen is an opaque procedure involving an unspecified number of blow-ups, making it prohibitive to compute all but the smallest examples. The construction due to Kapranov is much more explicit but requires a large number of blow-ups in increasing order of dimension along all projective subspaces spanned by certain sets of generic points. This, too, becomes very complicated, very quickly, and results in a projective embedding with extremely high codimension. Recently, Gillespie-Griffin-Levinson proved a conjecture of Monin-Rana concerning equations which define our moduli spaces of interest. These equations are given by all the 2×2 minors of certain matrices, for all i j in {1, 2, …, n-3}. In particular, the Monin-Rana equations yield a projective embedding with much lower codimension. Our work is to explicitly compare the three constructions given above and draw parallels where possible.

ABSTRACT

The moduli space of genus-zero curves has been widely studied by mathematicians due to its applicability to diverse research areas within mathematics as well as in theoretical physics. The goal of this research is to give a concrete and accessible presentation of recent results that describe the moduli space explicitly.
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