Can two projective surfaces intersect in points only?

Question

Let $S_1,S_2\subset \mathbb P^n$ be two algebraic surfaces (we may assume that $n=3$). Is it possible that $S_1\cap S_2=\{x_1,\dots,x_N\}$ a finite set of points?
I can imagine the surfaces two be disjoint, if they are parallel; to intersect along a curve, which should be the general situation; and to intersect along a surface, which occurs if $S_1=S_2$. Thus it seems like everything else is possible and that's why I am curious whether a point intersection could also be possible.

Answer 1

I am assuming the base field is algebraically closed.

You say rather casually in the question "we may assume that $n=3$", but the answer depends crucially on whether $n=3$ or not.

If $n=3$ then any two surfaces intersect in a set of dimension at least 1 (they cannot be disjoint, and there is no such thing as "parallel" in projective space). A reference is Hartshorne Theorem I.7.2.

If $n>3$ then certainly it is possible for two surfaces in $\mathbf P^n$ to intersect in a finite set of points --- for example, try coordinate planes in $\mathbf P^4$. Indeed, by the Theorem already mentioned, any two surfaces in $\mathbf P^4$ must have nonempty intersection, and in general the intersection will be a finite set.

If $n>4$ then any dimension up to 2 is possible for the intersection of two surfaces, but there are statements analogous to the above for intersections of higher-dimensional algebraic subsets.