TL;DR
This paper investigates the structure of certain Hilbert schemes, revealing the existence of singular lex-segment points on multiple components and extending these findings to related schemes like twisted cubics.
Contribution
It identifies and analyzes singular lex-segment points on various Hilbert schemes, highlighting their component structures and singularities, which was not previously documented.
Findings
Existence of singular lex-segment points on multiple irreducible components
Identification of singular points in related Hilbert schemes of curves
Extension of singular lex-segment point findings to twisted cubics
Abstract
We study the component structures of some standard-graded Hilbert schemes closely related to a Hilbert scheme of curves studied by Gotzmann. In particular, we encounter examples of singular lex-segment points lying on two and three irreducible components. We find further singular lex-segment points at nearby Hilbert schemes. We conclude by showing that the analogous example at the Hilbert scheme of twisted cubics also has a singular lex-segment point.
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Taxonomy
TopicsAlgebraic Geometry and Number Theory · Commutative Algebra and Its Applications · Algebraic structures and combinatorial models