A Study of Singularities on Rational Curves Via Syzygies

Content Provider	Semantic Scholar
Author	Cox, David Kustin, Andrew R. Polini, Claudia Ulrich, Bernd
Copyright Year	2011
Abstract	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
File Format	PDF HTM / HTML
DOI	10.1090/S0065-9266-2012-00674-5
Alternate Webpage(s)	http://people.math.sc.edu/kustin/papers/CoxKustinPolinUlrich-REVISED-SentTo-Memoirs-Jan-30-2012.pdf
Alternate Webpage(s)	http://people.math.sc.edu/kustin/papers/CoxKustinPolinUlrich-SentTo-Memoirs-and-arXiv-Feb-24-2011-REVISED-Nov-16-2011.pdf
Alternate Webpage(s)	https://arxiv.org/pdf/1102.5072v2.pdf
Alternate Webpage(s)	http://people.math.sc.edu/kustin/papers/CoxKustinPolinUlrich-SentTo-Memoirs-and-arXiv-Feb-24-2011.pdf
Alternate Webpage(s)	https://doi.org/10.1090/S0065-9266-2012-00674-5
Language	English
Access Restriction	Open
Content Type	Text
Resource Type	Article