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oai:docta.ucm.es:20.500.14352/575882023-08-28T15:01:37Zcom_20.500.14352_14col_20.500.14352_15

A bound for the arithmetic genus of curves in Grassmannians Giraldo Suárez, Luis 512.7 curve in the Grassmann variety arithmetic genus Castelnuovo’s bound Geometria algebraica 1201.01 Geometría Algebraica From the introduction: Let $X\subset\bbfP^n$ be a non-degenerate degree $d$ variety over the field of complex numbers, which is ruled by $k$-planes over a curve. Let us also suppose that there is no point of $X$ such that all the rules pass through it, i.e. that $X$ is not a cone.\par We can associate to $X$ a curve $C_X$ lying in the Grassmann variety of $k$-planes in $\bbfP^n$. The goal of this note is to show that the arithmetic genus of such a curve is bounded by $\pi(d,n)$, where $\pi(d,n)$ is Castelnuovo's bound for the genus of degree $d$ curves in $\bbfP^n$ .The proof of the result relies on a previous one that establishes that the bound holds and is sharp for ruled surfaces in $\bbfP^n$. The key idea of the proof is to show that the curve in $G(k,\bbfP^n)$ associated to $X$ spans at least a $\bbfP^n$. DGES Depto. de Álgebra, Geometría y Topología Fac. de Ciencias Matemáticas TRUE pub 2023-06-20T16:59:39Z 2023-06-20T16:59:39Z 2000 journal article https://hdl.handle.net/20.500.14352/57588 0933-7741 10.1515/form.2000.022 PB96-0659 metadata only access WALTER DE GRUYTER