Projective duality identifies the moduli space Bn parametrizing configurations of n points in P2 with the moduli space X(3,n) parametrizing configurations of n lines in P2. When considering degenerations of such objects, it is interesting to compare different compactifications of the above moduli spaces. In this work, we consider Gerritzen-Piwek's compactification \barBn and Kapranov's Chow quotient compactification \barX(3,n), and we show they are isomorphic. We construct and study an alternative compactification of Bn which parametrizes all possible n-pointed degenerate central fibers of the Mustafin joins of stable lattices associated to one-parameter degenerations of n points in P2. This corrects some of the results in Gerritzen-Piwek's paper. This is joint work in progress with Jenia Tevelev.
Tropical geometry provides a new set of purely combinatorial tools, which has been used to approach classical problems. In tropical geometry most algebraic computations are done on the classical side - using the algebra of the original variety. The theory developed so far has explored the geometric aspect of tropical varieties as opposed to the underlying (semiring) algebra and there are still many commutative algebra tools and notions without a tropical analogue. In the recent years, there has been a lot of effort dedicated to developing the necessary tools for commutative algebra using different frameworks, among which prime congruences, tropical ideals, tropical schemes. These approaches allows for the exploration of the properties of tropicalized spaces without tying them up to the original varieties and working with geometric structures inherently defined in characteristic one (that is, additively idempotent) semifields. In this talk we explore the relationship between tropical ideals and congruences to conclude that the variety of a prime tropical ideal is either empty or consists of a single point. This is joint work with D. Joó.
The interactions between geometric modeling, algebraic geometry, and commutative algebra are rich and varied. This talk will focus on the geometry of algebraic modeling, including practical applications, a bit of history, and some toric geometry and algebraic statistics.