Obstructions to deforming curves on a prime Fano 3‐fold

Hirokazu Nasu

Mathematische Nachrichten, 2019, vol. 292, issue 8, 1777-1790

Abstract: We prove that for every smooth prime Fano 3‐fold V, the Hilbert scheme HilbscV of smooth connected curves on V contains a generically non‐reduced irreducible component of Mumford type. We also study the deformations of degenerate curves C in V, i.e., curves C contained in a smooth anticanonical member S∈|−KV| of V. We give a sufficient condition for C to be stably degenerate, i.e., every small (and global) deformation of C in V is contained in a deformation of S in V. As a result, by using the Hilbert‐flag scheme of V, we determine the dimension and the smoothness of HilbscV at the point [C], assuming that the class of C in PicS is generated by h:=−KV|S together with the class of a line, or a conic on V.

Date: 2019
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https://doi.org/10.1002/mana.201800185

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