Linear syzygies and linear subspaces whose lines are multisecant
Abstract
The locus Sd(X) of d-secant lines to a projective variety X plays an important role in the study of projective varieties via projections (Lazarsfeld 1987, Kwak 1998, Beheshti-Eisenbud 2010). For a projective scheme X⊆ℙr satisfying Nd,2, we show that Sd(X)∪ X is cut out set-theoretically by the r-minors of a matrix M constructed from d-forms scheme-theoretically defining X and their linear syzygies. In particular, Sd(X)=ℙr if and only if every r-minor of M vanishes. Using this, we find a singular threefold X⊆ℙ5 with S4(X)≠ ℙ5 and (IX)3=0, and a normal fourfold X⊆ℙ7 with S3(X)≠ ℙ7 and (IX)2=0. The nonexistence of such varieties in the smooth case is an open question raised by the second author and by Gruson-Peskine. Next, we consider the locus Sk,d(X) of k-planes L such that L∩ X contains a hypersurface of degree ≥ d in L so that S1,d(X)=Sd(X). The locus Sk,d(X)∪ X is set-theoretically defined by the (r+1−k)-minors of M. As a result, we obtain determinantal equations vanishing on the (q+1)-secant variety σq+1X from the equations of σqX and their linear syzygies when σqX satisfies Nq+1,2. Finally, we extend the d-secant lemma for lines to the locus Sk,d(X) of k-planes. This yields a lower bound on the number of linear syzygies.
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BibTeX
@misc{han2026linear,
title = {{Linear syzygies and linear subspaces whose lines are multisecant}},
author = {Jong In Han and Sijong Kwak},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.12078},
eprint = {2610.12078},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.12078},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Linear syzygies and linear subspaces whose lines are multisecant
AU - Jong In Han
AU - Sijong Kwak
PY - 2026
DA - 2026/10/08
PB - arXiv
DO - 10.48550/arxiv.2610.12078
UR - https://arxiv.org/abs/2610.12078
AB - The locus S_d(X) of d-secant lines to a projective variety X plays an important role in the study of projective varieties via projections (Lazarsfeld 1987, Kwak 1998, Beheshti-Eisenbud 2010). For a projective scheme X⊆ℙ^r satisfying N_d,2, we show that S_d(X)∪ X is cut out set-theoretically by the r-minors of a matrix M constructed from d-forms scheme-theoretically defining X and their linear syzygies. In particular, S_d(X)=ℙ^r if and only if every r-minor of M vanishes. Using this, we find a singular threefold X⊆ℙ^5 with S_4(X)≠ ℙ^5 and (I_X)_3=0, and a normal fourfold X⊆ℙ^7 with S_3(X)≠ ℙ^7 and (I_X)_2=0. The nonexistence of such varieties in the smooth case is an open question raised by the second author and by Gruson-Peskine. Next, we consider the locus S_k,d(X) of k-planes L such that L∩ X contains a hypersurface of degree ≥ d in L so that S_1,d(X)=S_d(X). The locus S_k,d(X)∪ X is set-theoretically defined by the (r+1−k)-minors of M. As a result, we obtain determinantal equations vanishing on the (q+1)-secant variety σ_q+1X from the equations of σ_qX and their linear syzygies when σ_qX satisfies N_q+1,2. Finally, we extend the d-secant lemma for lines to the locus S_k,d(X) of k-planes. This yields a lower bound on the number of linear syzygies.
ER -