Here we classify$J$-embeddable surfaces, i.e. surfaces whose secant varieties have dimension at most 4, when the surfaces have two components at most.

Three-fold quasi-homogeneous isolated rational singularity is argued to define a four dimensional N = 2 SCFT. The SeibergWitten geometry is built on the mini-versal deformation of the singularity. We argue in this paper that the corresponding SeibergWitten differential is given by the GelfandLeray form of K. Saitos primitive form. Our result also extends the SeibergWitten solution to include irrelevant deformations.

We shall give a Counting Formula for the number of Fourier-Mukai partners of a K3 surface and consider three applications.

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