Prinzip der Erhaltung der Anzahl in Algebraic Geometry

"Principle of conservation of number" (PCN) is the translation of the title of this post.
This is a sequel of the post from last week (if you scroll down, you'll find it), in which I talked about the 15th Hilbert problem on the foundations of intersection theory in algebraic geometry.
First let me mention two examples (1 and 2) as to  how such a principle was put into practice and one counter example (3) where the distinction between a Weil and Cartier divisor is crucial.

(1) Since a plane curve of degree d can be deformed into a union of d lines, then the number of points in the intersection of two plane curves of degree d and d' (if finite) should be the product dd'. (Bezout Theorem) Here we assumed the intersection number is preserved.

(2) Question: how many lines in 3 dimensional space intersect 4 given skew lines?
If we assume the solution to this problem is finite and preserved under small perturbations of the lines, then we can move two of the lines such that they intersect in a point P. The solution to our problem in this case is clear: we want the number of lines through P which intersect the two other lines, this number is 2.
Now by the PCN this number 2 is preserved all the way through, hence the solution to the original problem must be 2 also.

(3) Now consider the quadric cone in projective 3 dimensional space given by the equation x^2+y^2=z^2. The line (Weil divisor) given by y=0 and x=z intersects the following family (parameter t) of conics t.w=x at a single point so long the parameter t is not zero. However, if the parameter t is zero, then this intersection consists of two points! Thus the intersection number is NOT preserved as we vary the parameter t.

The way I see it, these examples show that by using the PCN sometimes we get the right answer, but sometimes we just don't. Over the year and with problems much more sophisticated, it was extremely difficult to tell whether a problem on this matter was right or wrong.

Coming up next week one of my favorites theorems on classical algebraic geometry due to Fano.

The Italian School of Algebraic Geometry

People always refer to the Italians as great algebraic geometers. ``The Italian School of algebraic geometry"  it is referred to. I decided to do a little research with Professors and my library on the names and the kind of results that they got back at the begin of the 20th century.
Names that get into this are: Cremona, Severi, Eniques, Segre, Castelnuovo, Fano, Veronese, Bertini.
Partly the reason I got interested in this (not to mention I like algebraic geometry) is the following.

In 1900 David Hilbert proposed a list of guiding problems in mathematics. The 15th of such problems roughly speaking calls for rigorous foundations for intersection theory in algebraic geometry. Back then, Severi (who was a major figure in the community) supported Schubert's point of view on the subject. That is, trying to make rigorous Schubert's work. One of the tools Schubert used was for example the so-called ``Principle of conservation of number" (PCN) which turned out to be wrong (as we will show by example next time).

Coming up next week, the sequel of this post will be about examples for which the PCN goes wrong.