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Fixed point distribution on Hilbert scheme of points (2306.11521v2)

Published 20 Jun 2023 in math.AG, math-ph, and math.MP

Abstract: Let $\mathbf{k}$ be a closed field of characteristic zero. We prove that all monomial ideals sit in the curvilinear component of the Hilbert scheme of points of the affine space $\mathbb{A}_{\mathbf{k}}n$, answering a long-standing question about the distribution of torus-fixed points among punctual components. This result confirms that the punctual Hilbert scheme is connected, a property previously established only for the full Hilbert scheme in 1966 by Hartshorne.

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