We introduce and study the space curves of “h-extremal type”, that are curves of degreed and arithmetic genusg whose Rao function agrees, in a suitable interval depending ond andh, with the one of the “h-extremal” curves introduced by NotariSabadini. Our study is motivated by the literature of the last years concerning curves with large cohomology and their relations with the Hilbert scheme. p] Our main result is a Structure Theorem which gives some geometrical characterizations of such curves. The most intriguing is that ifd is sufficiently large with respect toh, a curve of hextremal type contains a planar subcurve of degreed —h and lies on a non integral quadric. As a consequence we can determine all possible Rao functions (for fixedd, g, h). We add several examples which show, in particular, that our result is the best possible forh ≤ 5.

中文翻译：

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包含平面子曲线的空间曲线的特征

我们引入和研究的“空间曲线ħ -extremal类型”，即是度的曲线d和算术属克其饶功能一致，在适当的时间间隔取决于d和ħ，用“的一个ħ -extremal” NotariSabadini引入的曲线。我们的研究是受最近几年有关具有大同调性的曲线及其与希尔伯特方案的关系的文献的启发。[p]我们的主要结果是一个结构定理，该定理给出了此类曲线的一些几何特征。最吸引人的是，如果d相对于h足够大，则十六进制类型的曲线包含一个度数为0的平面子曲线d — h，位于非整数二次曲面上。结果，我们可以确定所有可能的Rao函数（对于固定的d，g，h）。我们添加几个例子，说明，特别是，我们的结果是最好的^ h ≤5。