--Worksheet 8, Mathematics 535, Fall 2002




--Singular locus, Degree


--Goals: computing singular locus of projective varieties, degree of a projective variety


R=QQ[x_0..x_3]   -- work in P^3
V=Proj(R/ideal(x_0^2+x_1^2+x_2^2))   --a cone of rank 3 in P^3
singularLocus V
degree V
dim V

W=Proj(R/ideal(x_0^3+x_1^3+x_2^3+x_3^3)) -- another projective variety
ideal(V)+ideal(W)
I=radical oo
M=Proj(R/I)
degree M  -- note example of Bezout's theorem

romanideal=ideal(x_1^2*x_2^2+x_2^2*x_3^2+x_1^2*x_3^2-x_0*x_1*x_2*x_3)
singularLocus Proj(R/romanideal)
decompose ideal oo   -- the singular points on the Rman surface form 3 lines
degree romanideal    -- the Roman surface is a quartic hypersurface in P^3