Fat base schemes and unexpected hypersurfaces of type (d+k,d)
Abstract
We study hypersurfaces of degree d+k containing a prescribed zero-dimensional fat point scheme Z⊂ℙn and having multiplicity at least d along a general codimension-two linear subspace. We associate with this interpolation problem a vector bundle on ℙ1 whose splitting type determines the dimensions of all linear systems of type (d+k,d). For a fat point of multiplicity m, we determine the exact number of conditions detected by this bundle and introduce the corresponding effective length. This leads to an excess formula separating the contributions of the splitting type, the part of the fat structure invisible to degree-k fibrewise sections, and the Hilbert-function defect of the fixed scheme. We also prove a truncation theorem showing that multiplicities larger than k contribute only a uniform twist of the bundle and, geometrically, fixed hyperplane components. Finally, we exhibit explicit planar and three-dimensional examples showing that genuinely unexpected phenomena persist for nonreduced fixed base schemes, including geometrically irreducible unexpected hypersurfaces.
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BibTeX
@misc{malara2026base,
title = {{Fat base schemes and unexpected hypersurfaces of type $(d+k,d)$}},
author = {Grzegorz Malara},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.04564},
eprint = {2611.04564},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.04564},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Fat base schemes and unexpected hypersurfaces of type (d+k,d)
AU - Grzegorz Malara
PY - 2026
DA - 2026/11/03
PB - arXiv
DO - 10.48550/arxiv.2611.04564
UR - https://arxiv.org/abs/2611.04564
AB - We study hypersurfaces of degree d+k containing a prescribed zero-dimensional fat point scheme Z⊂ℙ^n and having multiplicity at least d along a general codimension-two linear subspace. We associate with this interpolation problem a vector bundle on ℙ^1 whose splitting type determines the dimensions of all linear systems of type (d+k,d). For a fat point of multiplicity m, we determine the exact number of conditions detected by this bundle and introduce the corresponding effective length. This leads to an excess formula separating the contributions of the splitting type, the part of the fat structure invisible to degree-k fibrewise sections, and the Hilbert-function defect of the fixed scheme. We also prove a truncation theorem showing that multiplicities larger than k contribute only a uniform twist of the bundle and, geometrically, fixed hyperplane components. Finally, we exhibit explicit planar and three-dimensional examples showing that genuinely unexpected phenomena persist for nonreduced fixed base schemes, including geometrically irreducible unexpected hypersurfaces.
ER -