On parameterizations of plane rational curves and their syzygies (Q2805373)

The paper under review describes a relationship between the geometry of a rational plane curve \(C\subset\mathbb P^2\) of degree \(d\) and its splitting type. Such a curve \(C\) is the image of a generically injective map \(f:\mathbb P^1\to\mathbb P^2\) given by polynomials \(f_0, f_1, f_2 \in S_d\), where \(S\) is the homogeneous coordinate ring for \(\mathbb P^1\). Setting \(I =(f_0,f_1,f_2)\subset S\), the kernel of the natural map \(S(-d)^3\to I\) has the form \(S(-k-d)\oplus S(k-2d)\), which defines the \textit{splitting type} \((k,d-k)\). For \(C\) general, the splitting type is \((\lfloor d/2 \rfloor, \lceil d/2 \rceil)\). \textit{M. G. Ascenzi} showed that if \(C\) has a singularity of multiplicity \(m \geq (d-1)/2\), then the splitting type is \((m,d-m)\) [Commun. Algebra 16, No. 11, 2193--2208 (1988; Zbl 0675.14010)]: moreover if \(C\) has splitting type \((1,d-1)\), then \(C\) must have a point of multiplicity \(d-1\); if \(C\) has splitting type \((2,d-2)\), then either \(C\) has a point of multiplicity \(d-2\) or \(C\) is a nodal projection of a smooth curve lying on a quadric surface in \(\mathbb P^3\).NEWLINENEWLINEThe authors extend these classifications by proving their main theorem: A rational curve \(C \subset \mathbb P^2\) has splitting type \((k,d-k)\) with \(1 \leq k \leq d/2\) if and only if \(C\) is the projection of a rational curve \(D \subset \mathbb P^{k+1}\) lying on a rational normal scroll \(S\) from \(k-1\) points outside of \(S\). They first construct \(D^\prime \subset \mathbb P^2 \times \mathbb P^1\) as the graph of the map \(f\) and use the lowest degree form in one of the bi-graded pieces of the ideal to construct \(S^\prime\), using partial derivatives to show that both are smooth. Then they interpret \(S^\prime\) as a Hirzebruch surface and map it into \(\mathbb P^{k+1}\) using a suitable linear system and let \(D \subset S\) be the corresponding images. Their method shows that if \(S\) is smooth, then \(C\) has no point of multiplicity \(m \geq (d-1)/2\); otherwise \(S\) is a cone and \(D\) passes through the vertex with multiplicity \(d-k\), when \(C\) has a point of multiplicity \(d-k\).