충청수학회지 (Journal of the Chungcheong Mathematical Society)

  • 제37권2호
  • /
  • Pages.57-66
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  • 2024
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  • 1226-3524(pISSN)
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  • 2383-6245(eISSN)
충청수학회 (The Chungcheong Mathematical Society)
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SOME GEOMETRIC PROPERTIES OF GOTZMANN COEFFICIENTS

  • Jeaman Ahn (Department of Mathematics Education Kongju National University)
  • 투고 : 2024.01.28
  • 심사 : 2024.04.29
  • 발행 : 2024.05.31
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초록

In this paper, we study how the Hilbert polynomial, associated with a reduced closed subscheme X of codimension 2 in ℙN, reveals geometric information about X. Although it is known that the Hilbert polynomial can tell us about the scheme's degree and arithmetic genus, we find additional geometric information it can provide for smooth varieties of codimension 2. To do this, we introduce the concept of Gotzmann coefficients, which helps to extract more information from the Hilbert polynomial. These coefficients are based on the binomial expansion of values of the Hilbert function. Our method involves combining techniques from initial ideals and partial elimination ideals in a novel way. We show how these coefficients can determine the degree of certain geometric features, such as the singular locus appearing in a generic projection, for smooth varieties of codimension 2.

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참고문헌

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