{"product_id":"a-study-of-singularities-on-rational-curves-via-syzygies","title":"A Study of Singularities on Rational Curves Via Syzygies","description":"\u003cp\u003eConsider a rational projective curve $\\mathcal{C}$ of degree $d$ over an algebraically closed field $\\pmb k$. There are $n$ homogeneous forms $g_{1},\\dots, g_{n}$ of degree $d$ in $B=\\pmb k[x, y]$ which parameterize $\\mathcal{C}$ in a birational, base point free, manner. The authors study the singularities of $\\mathcal{C}$ by studying a Hilbert-Burch matrix $\\varphi$ for the row vector $[g_{1},\\dots, g_{n}]$. In the ``General Lemma'' the authors use the generalized row ideals of $\\varphi$ to identify the singular points on $\\mathcal{C}$, 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 $\\mathcal{C}$ which corresponds to a generalized zero of $\\varphi$. In the `'triple Lemma'' the authors give a matrix $\\varphi'$ whose maximal minors parameterize the closure, in $\\mathbb{P}^{2}$, of the blow-up at $p$ of $\\mathcal{C}$ in a neighborhood of $p$. The authors apply the General Lemma to $\\varphi'$ in order to learn about the singularities of $\\mathcal{C}$ in the first neighborhood of $p$. If $\\mathcal{C}$ has even degree $d=2c$ and the multiplicity of $\\mathcal{C}$ at $p$ is equal to $c$, then he applies the Triple Lemma again to learn about the singularities of $\\mathcal{C}$ in the second neighborhood of $p$. Consider rational plane curves $\\mathcal{C}$ of even degree $d=2c$. The authors classify curves according to the configuration of multiplicity $c$ singularities on or infinitely near $\\mathcal{C}$. There are $7$ possible configurations of such singularities. They 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 $\\mathcal{C}$ of degree $2c$ is equivalent to the study of the scheme of generalized zeros of the fixed balanced Hilbert-Burch matrix $\\varphi$ for a parameterization of $\\mathcal{C}$.\u003c\/p\u003e\u003cul\u003e\n\u003cli\u003e\n\u003cb\u003eAuthor:\u003c\/b\u003e David A. Cox, Andrew R. Kustin, Claudia Polini, Bernd Ulrich\u003c\/li\u003e\n\u003cli\u003e\n\u003cb\u003ePublisher:\u003c\/b\u003e American Mathematical Soc.\u003c\/li\u003e\n\u003cli\u003e\n\u003cb\u003ePublished:\u003c\/b\u003e 2013-02-26\u003c\/li\u003e\n\u003cli\u003e\n\u003cb\u003ePages:\u003c\/b\u003e 132\u003c\/li\u003e\n\u003c\/ul\u003e","brand":"American Mathematical Soc.","offers":[{"title":"Default Title","offer_id":62808707137907,"sku":"9780821887431","price":200.0,"currency_code":"USD","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0987\/0812\/8115\/files\/content_94e75414-965d-4433-9050-01d19cdc7e51.jpg?v=1777969149","url":"https:\/\/readaura.store\/products\/a-study-of-singularities-on-rational-curves-via-syzygies","provider":"Read Aura","version":"1.0","type":"link"}