On the Geometric Interpretation of Primes via Singularity Theory

This paper proposes a novel framework for interpreting the arithmetic behavior of prime numbers using tools from singularity theory and algebraic geometry. By modeling primes as geometric objects arising from the resolution of singularities on arithmetic surfaces, we investigate how the structure of singular points reflects arithmetic obstructions and cohomological conditions. In particular, we explore how blowups, exceptional divisors, and Milnor numbers provide a geometric language for understanding prime gaps, congruence conditions, and density. The approach also connects with étale cohomology and the behavior of L-functions under local deformation. Through this reinterpretation, we propose a categorification of prime arithmetic via the geometry of singular points, potentially paving the way for new insights into global number-theoretic phenomena through local geometric data.