This is presentation of my Ph.D. thesis which is related to enumerative geometry and moduli spaces.
The inspiration for this project comes from enumeration of rational curves of given degree in the projective plane.
The problem of enumeration of rational goes back to 19th century, and was solved by Kontsevich and Manin in 1994.
The first solution was based on the development of the theory of moduli spaces, in particular $\overline{M}_{0,n}$
the compact algebraic space of rational nodal stable curves with $n$ marked points.
The unachieved goal was to develop a similar theory for enumeration of surfaces in 3-dimensional projective space.
The first step would be to find a smooth compactification for a space of generic configurations of $n$ marked points.
There is a construction of Kapranov which he called "Chow quotients" of Grassmanians which generalizes
$\overline{M}_{0,n}$ to any dimension, in particular it creates a similar space for configurations of $n$ marked points
In the projective plane; it has several nice properties, but it is not smooth even for 6 points in the plane
(so it would be hard to talk about intersection theory). Here, a new version of Kapranov's construction is presented,
by a similar technology but with a blow-up idea: we add lines connecting pairs of marked points before applying Chow quotients.
We prove that the new space for configuration of 6 marked points in the plane is smooth.
Another result (joint with S. Carmeli), which is obtained by intersection theory, is the enumeration of surfaces of given degree
with a singular line, vanishing to order k at the line.
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