This is a course not only about intersection theory but intended to introduce modern language of algebraic geometry and build up tools for solving
Let S = Spec(A)
An algebraic surface is said to be of degree , where is the maximum sum of powers of all terms
In characteristic zero, singularities can be resolved by
More generally, every irreducible cubic surface (possibly singular) over an algebraically closed field is rational unless it is the projective cone over a cubic curve
The surface X~ X ~ is called a weak del Pezzo surface, which is still blowup of 6 points on P2 P 2, but in less general positions, so H2(X~) = Z7 H 2 ( X ~) = Z 7
The proof claims : "I have to prove that a multiple line is impossible
Let n be a positive integer and let A be the subring of C[x, y] generated by x, xy,, xyn
If an internal link led you here, you may wish to change the link to point directly to the intended article
If the defining three-variate function is a polynomial, the surface is
2)
This should be possible to show by hand, since the possibilities for singular cubic surfaces are not so many
The number of singular points is bounded by the arithmetic genus (given by your adjunction formula) for irreducible curves in the family
Let X be a smooth algebraic variety or smooth complex manifold and Y be a smooth subvariety of X
We want a surface to have the plane Cusp{y2=x3,z=0}as a singular locus
3
Consider the Kummer surface K K associated to X X, that is the quotient of X X by the action of involution on X X, x ↦ −x x ↦ − x
This gives a map $\mathbb P^{n-1}\rightarrow H$, the cubic hypersurface, which is birational
Let k be an algebraically closed field
There is a polynomial Δ, in the coefficients of F, which vanishes precisely when the projective hypersurface F = 0 is singular
Optional complement
Let H and E are two irreducible components of the singular locus of X i