Understanding the structure of affine algebraic varieties and its applications

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I have studied the structure of affine algebraic surfaces. I gave another proof of the classification of the additive group scheme action on the affine plane and proved that the logarithmic 12th genus of the smooth part of a normal affine surface of logarithmic Kodaira dimension 1 is positive. Moreover, by using some recent results on affine algebraic surfaces, I studied curves on a normal projective rational surface whose complements have non-positive Euler characteristic and gave an upper bound for the number of singular points of a normal del Pezzo surface of Picard number 1 with only rational log canonical singularities. I also gave necessary and sufficient conditions for a polynomial in the polynomial ring over a unique factorization domain to be closed.