平面中三生成理想的几何
Geometry of three--generated ideals in the plane
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中文总结 AI 辅助
本文研究平面中由三个同次形式极小生成的齐次理想参数空间的几何，通过分层、普适族构造及合冲丛半稳定性，证明光滑性、不可约性并给出数值限制，特别完整刻画了三次形式情形。
中文摘要 AI 辅助
我们研究了由三个同次形式极小生成的 k[x,y,z] 中齐次理想的参数空间的几何，这些空间按其基方案的次数和其合冲的初始次数进行分层。我们通过 Quot 和 Hilbert 概形实现这些层，构造 Bourbaki 概形的普适族，并证明自由轨迹的光滑性和不可约性。然后我们将这些空间与梯度理想和平面曲线的轨迹进行比较。利用相关合冲丛的斜率半稳定性，我们给出了精细的 du Plessis--Wall 界限的向量丛解释，并对任意三生成理想获得了相同的数值限制。最后，在三次形式三元组的情形下，我们确定了所有非空次数层，证明它们是光滑且不可约的，并根据约化四次平面曲线的 Bourbaki 分层识别了它们的梯度轨迹。
英文摘要
We study the geometry of parameter spaces of homogeneous ideals in k[x,y,z] minimally generated by three forms of the same degree, stratified by the degree of their base scheme and by the initial degree of their syzygies. We realize these strata through Quot and Hilbert schemes, construct universal families of Bourbaki schemes, and prove smoothness and irreducibility results for the free loci. We then compare these spaces with the loci of gradient ideals and plane curves. Using slope semistability of the associated syzygy bundles, we give a vector-bundle interpretation of the refined du Plessis--Wall bounds and obtain the same numerical restriction for arbitrary three-generated ideals. Finally, in the case of triples of cubic forms, we determine all nonempty degree strata, prove that they are smooth and irreducible, and identify their gradient loci in terms of the Bourbaki stratification of reduced quartic plane curves.
发表机构
- IMECC, University of Campinas (UNICAMP)（坎皮纳斯大学 IMECC）
- Université Bourgogne Europe, CNRS, IMB UMR 5584（勃艮第欧洲大学，法国国家科学研究中心，IMB）
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