ag.algebraic geometry - 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).