Quantitative Riemann-Roch: Bézout Bounds for Effective Divisors on Moduli Spaces

The Riemann-Roch theorem stands as a cornerstone in both complex analysis and algebraic geometry, offering profound insights into the dimensions of spaces of meromorphic functions with specified zeros and poles. While its classical formulation provides qualitative existence results, the evolving landscape of modern algebraic geometry, particularly the study of intricate spaces such as moduli spaces, necessitates a more quantitative approach. Moduli spaces, often singular and defined as algebraic stacks, present significant challenges to traditional intersection theory. This paper addresses these challenges by developing a framework for a "Quantitative Riemann-Roch" theorem. Our primary objective is to establish explicit Bézout bounds for effective divisors on various moduli spaces. We leverage advanced tools from intersection theory on stacks, cohomology theory, Chern classes, and the theory of virtual fundamental classes, which are essential for navigating the complex geometry of these spaces. By providing rigorous, quantitative estimates for intersection products of effective divisors, this work contributes to a deeper understanding of the geometric and topological invariants of moduli spaces. The derived bounds have significant implications for enumerative geometry, especially in the context of Gromov-Witten theory, and for the classification of algebraic varieties.