A nonstandard proof of continuity of affine varieties

TL;DR

This paper employs nonstandard analysis to demonstrate that affine varieties deform infinitesimally when their defining polynomials are infinitesimally deformed, extending classical polynomial root continuity results to higher-dimensional algebraic varieties.

Contribution

It introduces a nonstandard analysis approach to prove the infinitesimal deformation of affine varieties under polynomial perturbations, a novel extension of classical root continuity.

Findings

Affine varieties deform infinitesimally with polynomial perturbations

Nonstandard analysis provides a new proof technique for algebraic geometry

Classical root continuity results extend to higher-dimensional varieties

Abstract

Extending the classical result that the roots of a polynomial with coefficients in $\mathbf{C}$ are continuous functions of the coefficients of the polynomial, nonstandard analysis is used to prove that if $\mathcal{F} = \{f_{\lambda} :\lambda \in \Lambda\}$ is a set of polynomials in $\mathbf{C}[t_1,\ldots, t_n]$ and if $^*\mathcal{G} = \{g_{\lambda} :\lambda \in \Lambda\}$ is a set of polynomials in $^*\mathbf{C}_0[t_1,\ldots, t_n]$ such that $g_{\lambda}$ is an infinitesimal deformation of $f_{\lambda}$ for all $\lambda \in \Lambda$, then the nonstandard affine variety $^*V_0(\mathcal{G})$ is an infinitesimal deformation of the affine variety $V(\mathcal{F})$.