论文标题
脂肪点方案的Kähler差分模块的Hilbert多项式模块
Hilbert Polynomials of Kähler Differential Modules for Fat Point Schemes
论文作者
论文摘要
Given a fat point scheme $\mathbb{W}=m_1P_1+\cdots+m_sP_s$ in the projective $n$-space $\mathbb{P}^n$ over a field $K$ of characteristic zero, the modules of Kähler differential $k$-forms of its homogeneous coordinate ring contain useful information about algebraic and geometric properties of $ \ mathbb {w} $当$ k \ in \ {1，\ dots，n+1 \} $。在本文中，我们明确确定其Hilbert多项式的价值，以$ k = n+1 $，确认了早期的猜想。更确切地说，此值由脂肪点方案的多重性$ \ Mathbb {y} =（M_1-1）P_1 + \ CDOTS +（M_S-1）P_S $。对于$ n = 2 $，这使我们能够确定$ k = 1,2,3 $的KählerDindical$ k $ forms的Hilbert多项式，并为$ k = 2 $的规律性指数产生急剧的限制。
Given a fat point scheme $\mathbb{W}=m_1P_1+\cdots+m_sP_s$ in the projective $n$-space $\mathbb{P}^n$ over a field $K$ of characteristic zero, the modules of Kähler differential $k$-forms of its homogeneous coordinate ring contain useful information about algebraic and geometric properties of $\mathbb{W}$ when $k\in\{1,\dots, n+1\}$. In this paper we determine the value of its Hilbert polynomial explicitly for the case $k=n+1$, confirming an earlier conjecture. More precisely this value is given by the multiplicity of the fat point scheme $\mathbb{Y} = (m_1-1)P_1 + \cdots + (m_s-1)P_s$. For $n=2$, this allows us to determine the Hilbert polynomials of the modules of Kähler differential $k$-forms for $k=1,2,3$, and to produce a sharp bound for the regularity index for $k=2$.
