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If [1:0:....:0] is an s-fold singularity of a degree $r$ hypersurface $F$ in $\mathbb{P}^n$ then the hypersurface can be written as $F=x_0^{r-s}g_s(x_1,...,x_n) + x_0^{r-s-1}g_{s+1}(x_1,...,x_n) + ... …

If you are interested in complex manifolds I would recommend Complex Geometry: an Introduction by Huybrechts. I also think the Toric Varieties by Cox, Little, and Schenck is an excellent introducti …

Aravind Asok has an entire website devoted to pointing out resources for learning $\mathbf{A}^1$-homotopy theory. It is organized quite well. The concept list section of the page has lots of wikiped …

A function f: ($\mathbb{C}^n$,0) $\to$ ($\mathbb{C}$,0) is considered a hypersurface singularity if the point $(0,0,\dots,0)$ is the only point in the ideal $\langle \frac{\partial f}{\partial x_1}, \ …

Geometric invariant theory works well when the algebraic group $G$ acting on a variety is reductive. There has been recent work by Doran and Kirwan here and here to find a canonical method of constru …

I think the Clay Math books have nice descriptions about Landau-Ginzburg Models. The book called Mirror Symmetry and Dirichlet Branes and Mirror Symmetry both have nice physical and mathematical desc …