• Journal of the Korean Institute of Intelligent Systems
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• v.12 no.1
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• pp.90-95
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• 2002

In this paper, we show that there exists a one-to-one correspondence between the collection of generalized ideals on a set and the collection of smooth ideals on a set. We study some properties of the images and preimages of smooth ideals. Smooth ideals, Generalized ideals, Images and preimages of smooth ideals.

• Journal of the Chungcheong Mathematical Society
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• v.37 no.2
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• pp.57-66
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• 2024

In this paper, we study how the Hilbert polynomial, associated with a reduced closed subscheme X of codimension 2 in ℙ^N, 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.

• Bulletin of the Korean Mathematical Society
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• v.58 no.1
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• pp.113-132
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• 2021

We show that if a divisor centered over a point on a smooth surface computes a minimal log discrepancy, then the divisor also computes a log canonical threshold. To prove the result, we study the asymptotic log canonical threshold of the graded sequence of ideals associated to a divisor over a variety. We systematically study this invariant and prove a result describing which divisors compute asymptotic log canonical thresholds.