Singularities of plane rational curves via projections. (English) Zbl 1390.14183
In this note, the authors tackle a problem of determining singularities of a plane rational curve \(C\) from its parameterization \(\mathbf{f}=(f_0:f_1:f_2)\). In order to study the singularities of \(C\) of degree \(n\), they use the fact that the parameterization of \(C\) defines a projection \(\pi:\mathbb{P}^n\dashrightarrow\mathbb{P}^2\), which is generically one-to-one from the rational normal curve \(C_n\subset\mathbb{P}^n\) onto its image \(\pi(C_n)=C\subset\mathbb{P}^2\). Then they explore the secant varieties to \(C_n\). In particular, they define via \(\mathbf{f}\) certain \(0\)-dimensional schemes \(X_k\subset\mathbb{P}^k\), \(2\leq k\leq (n-1)\), which encode all information on the singularities of multiplicity \(\geq k\) of \(C\).
The authors give also a series of algorithms which allow one to obtain information about the singularities of \(C\) from such schemes \(X_k\). The algorithms presented in the paper in pseudo code can be easily implemented in symbolic algebra programs such as CoCoA, Macaulay2 or Bertini.
Editorial remark: in [A. Gimigliano and M. Idà, Geom. Dedicata 217, No. 1, Paper No. 5, 14 p. (2023; Zbl 1504.14057)], the last two authors give a counterexample to Lemma 4.2.
The authors give also a series of algorithms which allow one to obtain information about the singularities of \(C\) from such schemes \(X_k\). The algorithms presented in the paper in pseudo code can be easily implemented in symbolic algebra programs such as CoCoA, Macaulay2 or Bertini.
Editorial remark: in [A. Gimigliano and M. Idà, Geom. Dedicata 217, No. 1, Paper No. 5, 14 p. (2023; Zbl 1504.14057)], the last two authors give a counterexample to Lemma 4.2.
Reviewer: Justyna Szpond (Kraków)
MSC:
| 14Q05 | Computational aspects of algebraic curves |
| 68W30 | Symbolic computation and algebraic computation |
Citations:
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