Abstract
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.
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Greco, S., Paxia, G. Characterizations of space curves containing a planar subcurve. Collect. Math. 61, 131–149 (2010). https://doi.org/10.1007/BF03191238
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DOI: https://doi.org/10.1007/BF03191238