Abstract: The discriminant of a space of functions is the closed subset consisting

of the functions which are singular in some sense. Specifically, we will
consider the the space of non-singular sections of a very ample line
bundle L on a fixed non-singular variety. In this set-up, Vakil and Wood
proved a stabilization behaviour for the class of complements of
discriminants in the Grothendieck group of varieties. In this talk, I
will discuss a topological approach for obtaining the cohomological
counterpart of Vakil and Wood's result, which implies that the k-th
cohomology group of the space of non-singular sections remains the same
if one takes a sufficiently high power of the line bundle L. As an
application, I will discuss a result by my former PhD student Angelina
Zheng on the stabilization of the cohomology of the moduli space of
trigonal curves.