Let $C$ be a plane rational curve of degree $d$ and $p: ilde C ightarrow C$ its normalization. We are interested in the {it splitting type} $(a,b)$ of $C$, where $mathcal{O}_{mathbb{P}^1}(-a-d)oplus mathcal{O}_{mathbb{P}^1}(-b-d)$ gives the syzigies of the ideal $(f_0,f_1,f_2)subset K[s,t]$, and , $(f_0,f_1,f_2)$ is a parameterization of $C$. We want to describe in which cases $(a,b)=(k,d-k)$ ($2kleq d)$, via a geometric description; namely we show that $(a,b)=(k,d-k)$ if and only if $C$ is the projection of a rational curve on a rational normal surface in $PP^{k+1}$.

On parameterizations of plane rational curves and their syzygies / Bernardi, Alessandra; Gimigliano, A.; Idà, M.. - In: MATHEMATISCHE NACHRICHTEN. - ISSN 0025-584X. - 289:5-6(2016), pp. 537-545. [10.1002/mana.201500264]

On parameterizations of plane rational curves and their syzygies

Bernardi, Alessandra;
2016

Abstract

Let $C$ be a plane rational curve of degree $d$ and $p: ilde C ightarrow C$ its normalization. We are interested in the {it splitting type} $(a,b)$ of $C$, where $mathcal{O}_{mathbb{P}^1}(-a-d)oplus mathcal{O}_{mathbb{P}^1}(-b-d)$ gives the syzigies of the ideal $(f_0,f_1,f_2)subset K[s,t]$, and , $(f_0,f_1,f_2)$ is a parameterization of $C$. We want to describe in which cases $(a,b)=(k,d-k)$ ($2kleq d)$, via a geometric description; namely we show that $(a,b)=(k,d-k)$ if and only if $C$ is the projection of a rational curve on a rational normal surface in $PP^{k+1}$.
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Bernardi, Alessandra; Gimigliano, A.; Idà, M.
On parameterizations of plane rational curves and their syzygies / Bernardi, Alessandra; Gimigliano, A.; Idà, M.. - In: MATHEMATISCHE NACHRICHTEN. - ISSN 0025-584X. - 289:5-6(2016), pp. 537-545. [10.1002/mana.201500264]
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/11572/141738
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