Give an example of two irreducible subvarieties of \(\PP^3\) whose intersection is reducible.
Solution
Use homogeneous coordinates \[ [x:y:z:w] \] on \(\PP^3\) and put \[ X=V(w), \qquad Y=V(xy-zw). \]
1. The plane \(X\) is irreducible.
Its homogeneous coordinate ring is \[ \CC[x,y,z,w]/(w)\cong\CC[x,y,z], \] which is an integral domain.
2. The quadric surface \(Y\) is irreducible.
Consider \[ F=xy-zw \] as a polynomial of degree one in \(x\) over \[ A=\CC[y,z,w]. \] If \[ F=gh \] were a nontrivial factorization, then after interchanging factors one factor, say \(g\), would lie in \(A\), while \[ h=ux+v \] for some \(u,v\in A\). Comparing the coefficient of \(x\) and the constant term gives \[ gu=y, \qquad gv=-zw. \] Thus \(g\) divides both \(y\) and \(zw\). These are relatively prime in the UFD \(A\), so \(g\) is a unit. Hence \(F\) is irreducible.
Since \(\CC[x,y,z,w]\) is a UFD, the irreducible element \(F\) is prime. Therefore the homogeneous coordinate ring of \(Y\) is a domain, and \(Y\) is irreducible.
3. Their intersection is the reducible curve\[ \boxed{ X\cap Y = V(w,x)\cup V(w,y). } \]
On the plane \(w=0\), the quadric equation becomes \[ xy=0. \] Hence \[ X\cap Y = V(w,xy) = V(w,x)\cup V(w,y). \] The two closed subsets are distinct projective lines. Neither contains the other, so their union is reducible.
4. Q.E.D.
Steps <1>1–<1>2 give two irreducible subvarieties of \(\PP^3\), and step 3 proves that their intersection is reducible.