Existence of smoothing of Calabi-Yau cones over \$dP_{1}\$ and \$dP_{2}\$ - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T08:11:15Z http://mathoverflow.net/feeds/question/12962 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/12962/existence-of-smoothing-of-calabi-yau-cones-over-dp-1-and-dp-2 Existence of smoothing of Calabi-Yau cones over \$dP_{1}\$ and \$dP_{2}\$ ronan-c 2010-01-25T20:18:56Z 2010-01-25T23:05:48Z <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#12985 Answer by Dmitri for Existence of smoothing of Calabi-Yau cones over \$dP_{1}\$ and \$dP_{2}\$ Dmitri 2010-01-25T23:05:48Z 2010-01-25T23:05:48Z <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>