线性合冲与含多重割线的线性子空间
Linear syzygies and linear subspaces whose lines are multisecant
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中文总结 AI 辅助
本文研究射影簇的多重割线轨迹，证明满足N_{d,2}的射影簇的割线轨迹可由矩阵子式定义，构造了两类奇异射影簇，推广了割线引理并得到线性合冲数量的下界。
中文摘要 AI 辅助
射影簇X的d重割线轨迹S_d(X)在通过投影研究射影簇中发挥重要作用（Lazarsfeld 1987, Kwak 1998, Beheshti-Eisenbud 2010）。对于满足N_{d,2}的射影概型X⊆ℙ^r，我们证明S_d(X)∪X在集合论意义上由矩阵M的r阶子式截出，其中M由概型论定义X的d次型及其线性合冲构造而成。特别地，S_d(X)=ℙ^r当且仅当M的所有r阶子式均为零。利用这一结论，我们找到奇异三维簇X⊆ℙ^5满足S_4(X)≠ℙ^5且(I_X)_3=0，以及正规四维簇X⊆ℙ^7满足S_3(X)≠ℙ^7且(I_X)_2=0；这类簇在光滑情形下是否存在是第二作者与Gruson-Peskine提出的公开问题。接下来，我们考虑k平面L的轨迹S_{k,d}(X)，使得L∩X包含L中次数≥d的超曲面，故S_{1,d}(X)=S_d(X)；S_{k,d}(X)∪X在集合论意义上由M的(r+1−k)阶子式定义。由此，当σ_qX满足N_{q+1,2}时，我们可从σ_qX的方程及其线性合冲得到在(q+1)重割线簇σ_{q+1}X上消失的行列式方程。最后，我们将直线的d重割线引理推广到k平面的轨迹S_{k,d}(X)，这给出了线性合冲数量的下界。
英文摘要
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 $X\subseteq\mathbb{P}^r$ satisfying $\textbf{N}_{d,2}$, we show that $S_d(X)\cup X$ 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, $S_d(X)=\mathbb{P}^r$ if and only if every $r$-minor of $M$ vanishes. Using this, we find a singular threefold $X\subseteq\mathbb{P}^5$ with $S_4(X)\ne \mathbb{P}^5$ and $(I_X)_3=0$, and a normal fourfold $X\subseteq\mathbb{P}^7$ with $S_3(X)\ne \mathbb{P}^7$ 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 $L\cap X$ contains a hypersurface of degree $\ge d$ in $L$ so that $S_{1,d}(X)=S_d(X)$. The locus $S_{k,d}(X)\cup X$ 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 $σ_{q+1}X$ from the equations of $σ_qX$ and their linear syzygies when $σ_qX$ satisfies $\textbf{N}_{q+1,2}$. 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.
发表机构
- Korea Institute for Advanced Study (KIAS)（韩国高等研究院）
- Korea Advanced Institute of Science and Technology (KAIST)（韩国科学技术院）
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