We consider a finite analytic morphism <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\varphi =(f,g)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>φ</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mi>f</mi> <mo>,</mo> <mi>g</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> defined from a complex analytic normal surface (Z, z) to <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\mathbb {C}^2$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">C</mi> </mrow> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation>. We describe the topology of the image by <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\varphi $</EquationSource> <EquationSource Format="MATHML"><math> <mi>φ</mi> </math></EquationSource> </InlineEquation> of a reduced curve on (Z, z) by means of iterated pencils defined recursively for each branch of the curve from the initial one <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\langle f,g \rangle $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">⟨</mo> <mi>f</mi> <mo>,</mo> <mi>g</mi> <mo stretchy="false">⟩</mo> </mrow> </math></EquationSource> </InlineEquation>. This result generalizes the one obtained in a previous paper for the case in which (Z, z) is smooth and the curve irreducible. The methods we use also permit us to describe the topological type of the discriminant curve of <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\varphi $</EquationSource> <EquationSource Format="MATHML"><math> <mi>φ</mi> </math></EquationSource> </InlineEquation>, in particular, the topological type of each branch of the discriminant can be obtained from the map without previous knowledge of the critical locus.

On the image of a curve in a normal surface by a plane projection

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2026年1月31日

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F. Delgado de la Mata,
H. Maugendre

摘要

We consider a finite analytic morphism $\varphi =(f,g)$ $φ = (f, g)$ defined from a complex analytic normal surface (Z, z) to $\mathbb {C}^2$ $C^{2}$ . We describe the topology of the image by $\varphi $ $φ$ of a reduced curve on (Z, z) by means of iterated pencils defined recursively for each branch of the curve from the initial one $\langle f,g \rangle $ $⟨ f, g ⟩$ . This result generalizes the one obtained in a previous paper for the case in which (Z, z) is smooth and the curve irreducible. The methods we use also permit us to describe the topological type of the discriminant curve of $\varphi $ $φ$ , in particular, the topological type of each branch of the discriminant can be obtained from the map without previous knowledge of the critical locus.