Specializations of families of curves of degree 4 in algebraic geometry - Piedra, Andrés (2026)
The objective of this review article is to synthesize and present, in a detailed manner, the technique of specialization of curves. Specifically, we review and analyze key examples on how certain curves of degree 4 and arithmetic genus 0 (parametrized by the Hilbert scheme Hilb4m+1(ℙ3)) can transform or degenerate into other curves. The computational calculation procedure is detailed and illustrated through the use of the Macaulay2 software. Finally, the utility of specializations for explicitly describing and classifying the irreducible components of Hilbert schemes and their corresponding Chow varieties is evaluated. The technique of specialization of families of curves has established itself as a powerful and rigorous method in algebraic geometry. By providing a well-defined stratification diagram, this tool allows for a precise classification of curves and an understanding of the connectivity of the components of the Hilbert scheme.