Matrix factorization and the generic plane projection of curve a singularity
Joan Elias
TL;DR
The paper connects matrix factorizations to the generic plane projections of irreducible curve singularities in $\mathbb{C}^n$, using the plane projection to produce a complete intersection $\mathcal{O}_{(Y,0)}=\mathbb{C}\{x,y\}/(F)$ and a MF $(A,B)$ with $AB=BA=F\,\mathrm{Id}$, yielding a finite, flat family of MFs that specializes to each fiber. It establishes a global MF in the flat family and analyzes how the associated minimal resolutions reflect the equisingularity of the projection. A principal contribution is a bijective correspondence between orbits of matrix factorizations (modulo $Gl_b$ actions) and isomorphism classes of $\mathcal{O}_{(Y,0)}$-algebras arising as $\mathcal{O}_{(X,0)}$, parameterized by the minimal number of generators $b=\beta_{\mathcal{O}_{(Y,0)}}(\mathcal{O}_{(X,0)})$, thereby linking MF data to the algebraic structure of the space curve and its generic plane projection. The work also provides a concrete monomial-curve example and develops a general framework to classify MF corresponding to a fixed plane curve singularity via $\operatorname{MatRep}_{F,b}$ and a $Gl_b$-action, yielding a thorough correspondence between MF representations and $\mathcal{O}_{(Y,0)}$-algebra saturations of the same projection.
Abstract
We study the matrix factorizations defined by the generic plane projections of a curve singularity of $\mathbb{C}^3$. On the other hand, given a plane curve singularity $Y\subset \mathbb{C}^2$ we study the family of matrix factorizations defined by the space curve singularities $X\subset \mathbb{C}^3$ such that $Y$ is the generic plane projection of $X$.