Existence of smoothing of Calabi-Yau cones over $dP_{1}$ and $dP_{2}$ - MathOverflow most recent 30 from http://mathoverflow.net2013-05-21T08:11:15Zhttp://mathoverflow.net/feeds/question/12962http://www.creativecommons.org/licenses/by-nc/2.5/rdfhttp://mathoverflow.net/questions/12962/existence-of-smoothing-of-calabi-yau-cones-over-dp-1-and-dp-2Existence of smoothing of Calabi-Yau cones over $dP_{1}$ and $dP_{2}$ronan-c2010-01-25T20:18:56Z2010-01-25T23:05:48Z
<p>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!</p>
http://mathoverflow.net/questions/12962/existence-of-smoothing-of-calabi-yau-cones-over-dp-1-and-dp-2/12985#12985Answer by Dmitri for Existence of smoothing of Calabi-Yau cones over $dP_{1}$ and $dP_{2}$Dmitri2010-01-25T23:05:48Z2010-01-25T23:05:48Z<p>The answer to this question is contained in the article of Mark Gross, page 33</p>
<p>Deforming Calabi-Yau Threefolds</p>
<p><a href="http://arxiv.org/abs/alg-geom/9506022" rel="nofollow">http://arxiv.org/abs/alg-geom/9506022</a> </p>
<p>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).</p>