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The blowdown of the zero section of the canonical bundle of the first del Pezzo surface $dP_{1}$, the blowup of $CP^{2}$ at one point, is a Calabi-Yau cone. I was just wondering if this cone admitted a smoothing. Is the same thing true for the blowdown of the zero section of the canonical bundle of the second del Pezzo surface $dP_{2}$, the blowup of $CP^{2}$ at two points. Thanks!

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up vote 3 down vote accepted

The answer to this question is contained in the article of Mark Gross, page 33

Deforming Calabi-Yau Threefolds

http://arxiv.org/abs/alg-geom/9506022

The first cone can not be smoothed the second one can be smoothed (in the terminology of Gross, which is standard, $dP_1$ is a del-Pezzo of degree 8, $dP_2$ is the del Pezzo of degree 7).

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