To a curve germ in d-space we can associate a numerical semigroup called its value semigroup. All numerical semigroups arise in this way. For a given semigroup we define its honest embedding dimension to be the minimal d for which there is a curve in d-space whose value semigroup is this semigroup. What are all the semigroups having honest embedding dimension d? Bresinsky answered this question for <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$d =2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>d</mi> <mo>=</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>, i.e. for the case of planar curves. The question for <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$d =3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>d</mi> <mo>=</mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation> (space curves) is open. Refining the question by the multiplicity m of the semigroup simplifies the work, in part because we must have <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$d \le m$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>d</mi> <mo>≤</mo> <mi>m</mi> </mrow> </math></EquationSource> </InlineEquation>. Our first Theorem classifies the numerical semigroups having multiplicity 4 and honest embedding dimension 3. Theorem <InternalRef RefID="FPar12">1</InternalRef> is a corollary of Theorem <InternalRef RefID="FPar13">2</InternalRef> which classifies the numerical semigroups for which <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$d = m$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>d</mi> <mo>=</mo> <mi>m</mi> </mrow> </math></EquationSource> </InlineEquation>, i.e those semigroups whose multiplicity m and honest embedding dimensional are equal. Our main tool is the Kunz cone, a convex polyhedral cone in <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$m-1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>m</mi> <mo>-</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> dimensions which allows visualization of the space of semigroups of multiplicity m. The answer provided in Theorem <InternalRef RefID="FPar13">2</InternalRef> is phrased as the intersection of the Kunz cone by another convex polyhedral cone.

The honest embedding dimension of a numerical semigroup

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2026年5月19日

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Richard Montgomery

摘要

To a curve germ in d-space we can associate a numerical semigroup called its value semigroup. All numerical semigroups arise in this way. For a given semigroup we define its honest embedding dimension to be the minimal d for which there is a curve in d-space whose value semigroup is this semigroup. What are all the semigroups having honest embedding dimension d? Bresinsky answered this question for $d =2$ $d = 2$ , i.e. for the case of planar curves. The question for $d =3$ $d = 3$ (space curves) is open. Refining the question by the multiplicity m of the semigroup simplifies the work, in part because we must have $d \le m$ $d \leq m$ . Our first Theorem classifies the numerical semigroups having multiplicity 4 and honest embedding dimension 3. Theorem 1 is a corollary of Theorem 2 which classifies the numerical semigroups for which $d = m$ $d = m$ , i.e those semigroups whose multiplicity m and honest embedding dimensional are equal. Our main tool is the Kunz cone, a convex polyhedral cone in $m-1$ $m - 1$ dimensions which allows visualization of the space of semigroups of multiplicity m. The answer provided in Theorem 2 is phrased as the intersection of the Kunz cone by another convex polyhedral cone.