What does this research mean for the field?

The very-compressed locus with Hilbert function (1,4,10,10) is the reduced support of a generically nonreduced irreducible component of the Hilbert scheme of 25 points on affine four-space over an algebraically closed field of characteristic zero. Novelty: ClaimNovelty.INCREMENTAL. Consensus alignment: ConsensusAlignment.NEUTRAL.

What question did this study set out to answer?

To establish whether the very-compressed locus with Hilbert function (1,4,10,10) forms the reduced support of a generically nonreduced irreducible component of the Hilbert scheme of 25 points on affine four-space.

September 7, 2026Open Access

A Generically Nonreduced Component for Hilbert Function (1,4,10,10)

View Full Paper

Ask AI

Bookmark

Authors

AFAlper FerudunMercer (Czechia)

Discussion

Member takes

Implication

Computational study demonstrates a generically nonreduced component for Hilbert function (1,4,10,10) in affine four-space, resolving the missing case in AIM Problem 31.

Key Points

To establish whether the very-compressed locus with Hilbert function (1,4,10,10) forms the reduced support of a generically nonreduced irreducible component of the Hilbert scheme of 25 points on affine four-space.
Calculated 244 primary-obstruction quadrics in 46 normal variables over the finite field F_3 using a patched msolve solver.
Conducted exact F4 Gröbner basis algorithm rounds to demonstrate that the normal obstruction algebra is zero-dimensional.
Transferred the computational certificate from characteristic 3 to characteristic zero via a properness argument combined with the Bialynicki-Birula criterion.
Identified that the locus is the reduced support of a generically nonreduced irreducible component of dimension 104, isomorphic to affine four-space times the Grassmannian Gr(10,20).
Supplied the omitted case a=10 for Hilbert functions (1,4,10,a), completing the classification of generic components across the entire interval in AIM Problem 31.

Cite This Study

Alper Ferudun (2026) studied this question.

synapsesocial.com/papers/6a9e85b3c3034f961570dffb https://doi.org/10.5281/zenodo.22328643

View Full Paper

Ask AI

Bookmark

Implication

Computational study demonstrates a generically nonreduced component for Hilbert function (1,4,10,10) in affine four-space, resolving the missing case in AIM Problem 31.

A Generically Nonreduced Component for Hilbert Function (1,4,10,10)

Discussion

Member takes

Implication

Key Points

Cite This Study

A Generically Nonreduced Component for Hilbert Function (1,4,10,10)

Discussion

Member takes

Implication

Key Points

Cite This Study

Also Consider

Also Consider