The moving image at the top of the mathematics 535 home page during the first week shows a portion of an affine real locus of the weighted projective space P(1,1,2). The weighted projective space can be imbedded in P^3 as the set of points {[a^2,ab,b^2,c^2]} so it is a cone over a plane conic. The equation of the weighted projective space in P^3 is z_0z_2=z_1^2. The picture shows a portion of the affine part of the surface where z_3=1. Drawn on the surface is the curve y^2=x_1^4-x_2^4, which is weighted homogeneous of degree 4 in x_1,x_2,y with weights (1,1,2). The affine curve y^2=x^4-1 is the affine portion of a smooth curve in P^3 which is the intersection of two quadrics. Thus, this projective space curve has degree 4 and since it is a complete intersection of degree 2 surfaces it has arithmetic genus 1. It is birational to the projective plane curve obtained by homogenizing y^2-x^4+1 with respect to a new variable x_3, but this gives a singular plane curve. The weighted projective spaces are examples of toric varieties. In this case the fan consists of all cones spanned by the rays determined by (1,0), (0,1), (-2,-1). The dual cones are drawn on the right.
To verify that this toric variety is the weighted projective space, consider the cones c_1 bounded by rays (1,0) and (0,1), c_2 bounded by (-2,-1) and (0,1) and the cone c_3 bounded by (-2,-1) and (1,0). Since both c_1 and c_3 are cones generated by a basis of the integer lattice, the corresponding affine toric variety is affine 2 space A^2, with coordinate algebras K[x_1,x_2] and K[x_1x_2^{-2},x_2^{-1}]. The second cone has dual cone generated by (-1,0) and (-1,2), and the semigroup of lattice points in this cone is generated by (-1,0),(-1,2) and (-1,1) so the corresponding algebra is K[x_1^{-1},x_1^{-1}x_2^2,x_1^{-1}x_2]=K[T,U,V]/(TU-V^2), the coordinate ring of the quotient of A^2 by the action (a,b) -> (-a,-b). The corresponding open sets in weighted projective space consists of of the sets of points [1,x_2,x_1], [a,b,1], [a,1,c] respectively. The intersection of the first and third is the set of points [1,x_2,x_1]=[1/x_2,1,x_1/x_2^2], which coincides with the identification of the open set corresponding to the common face, the ray genrated by (1,0). Similarly the intersection of the first and second is the set of points of form [a,b,1]=[1,b/a,1/a^2]=[1,x_2,x_1]. The coordinates on [a,b,1] are a^2=1/x_1, ab=x_2/x_1, and b^2=x_2^2/x_1, in agreement with the identifications on the toric variety.