We study the matrix factorizations defined by the generic plane projections of a curve singularity of <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$(\mathbb {C}^n,0)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <msup> <mrow> <mi mathvariant="double-struck">C</mi> </mrow> <mi>n</mi> </msup> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. On the other hand, given a plane curve singularity <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$(Y,0)\subset (\mathbb {C}^2,0)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow> <mo stretchy="false">(</mo> <mi>Y</mi> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> <mo>⊂</mo> <mrow> <mo stretchy="false">(</mo> <msup> <mrow> <mi mathvariant="double-struck">C</mi> </mrow> <mn>2</mn> </msup> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> we study the family of matrix factorizations defined by the space curve singularities <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$(X,0)\subset (\mathbb {C}^n,0)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> <mo>⊂</mo> <mrow> <mo stretchy="false">(</mo> <msup> <mrow> <mi mathvariant="double-struck">C</mi> </mrow> <mi>n</mi> </msup> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> such that (Y, 0) is the generic plane projection of (X, 0).

Matrix Factorization and the Generic Plane Projection of Curve a Singularity

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2025年7月22日

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Joan Elias

摘要

We study the matrix factorizations defined by the generic plane projections of a curve singularity of $(\mathbb {C}^n,0)$ $(C^{n}, 0)$ . On the other hand, given a plane curve singularity $(Y,0)\subset (\mathbb {C}^2,0)$ $(Y, 0) \subset (C^{2}, 0)$ we study the family of matrix factorizations defined by the space curve singularities $(X,0)\subset (\mathbb {C}^n,0)$ $(X, 0) \subset (C^{n}, 0)$ such that (Y, 0) is the generic plane projection of (X, 0).