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!



Existence of smoothing of Calabi-Yau cones over $dP_{1}$ and $dP_{2}$
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).


