A Study of Singularities on Rational Curves Via Syzygies
A Study of Singularities on Rational Curves Via Syzygies
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DOI：
10.1090/s0065-9266-2012-00674-5
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发表时间：
2011-02
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影响因子：
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通讯作者：
David A. Cox;A. Kustin;C. Polini;B. Ulrich
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文献类型：
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作者：
David A. Cox;A. Kustin;C. Polini;B. Ulrich
考虑在代数闭域k上的有理d次投影曲线C，有n种齐次形式g1；：：：；在B中d次的gn = kk(x;y)，它以一种二分的、无基点的方式参数化C。我们通过研究行向量（g1;::; gn）的Hilbert-Burch矩阵来研究C的奇异性。在“一般引理”中，我们利用'的广义行理想来确定C上的奇异点、它们的多重度、每个奇异点上的分支数以及每个分支的多重度。设p为参数化平面曲线C上的一个奇异点，它对应于'的广义零。在“三重引理”我们给一个矩阵的0未成年人最大的参数化关闭,在P 2中,爆破的P C P。我们的社区应用一般引理的0为了了解C的奇点在第一社区P。如果C甚至学位d = 2 C和C的多样性在P = C,然后应用三重引理再次了解C的奇点第二社区P。考虑合理的平面曲线C d = 2 C的学位。我们根据c上或无穷近处的多重c个奇点的构形对曲线进行分类，这样的奇点有七种可能的构形。我们对对应于每个构型的Hilbert-Burch矩阵进行分类。研究2c次的固定有理平面曲线c上或无穷近处的多重c个奇点等价于研究c的参数化的固定平衡Hilbert-Burch矩阵的广义零点格式
Consider a rational projective curve C of degree d over an algebraically closed field k. There are n homogeneous forms g1;:::;g n of degree d in B = kk(x;y) which parameterize C in a birational, base point free, manner. We study the singularities of C by studying a Hilbert-Burch matrix ' for the row vector (g1;:::;g n). In the "General Lemma" we use the generalized row ideals of ' to identify the singular points on C, their multiplicities, the number of branches at each singular point, and the multiplicity of each branch. Let p be a singular point on the parameterized planar curve C which corresponds to a general- ized zero of '. In the "Triple Lemma" we give a matrix ' 0 whose maximal minors parameterize the closure, in P 2 , of the blow-up at p of C in a neighborhood of p. We apply the General Lemma to ' 0 in order to learn about the singularities of C in the first neighborhood of p. If C has even degree d = 2c and the multiplicity of C at p is equal to c, then we apply the Triple Lemma again to learn about the singularities of C in the second neighborhood of p. Consider rational plane curves C of even degree d = 2c. We classify curves according to the configuration of multiplicity c singularities on or infinitely near C. There are 7 possible configurations of such singularities. We classify the Hilbert-Burch matrix which corresponds to each configuration. The study of multiplicity c singularities on, or infinitely near, a fixed rational plane curve C of degree 2c is equivalent to the study of the scheme of generalized zeros of the fixed balanced Hilbert-Burch matrix ' for a parameterization of C. Let