What is the idea behind deformation to the normal cone and what are examples of its applications?



deformation to the normal cone
Here's a place to see the normal cone side-by-side with other familiar constructions, that I learned from Fulton's "Intersection Theory". Here $X \subset Y$.


Start with the space $Y \times {\mathbb P}^1$, thought of as a trivial family over ${\mathbb P}^1$. Blow up the subscheme $X \times \infty$. Now we still have a flat family over ${\mathbb P}^1$, in which all the fibers except the one over $\infty$ are still copies of $Y$. The fiber over $\infty$ is reducible: one piece $Z\_1$ is the blowup of $Y$ along $X$, and the other is the projective completion $Z\_2$ of the normal cone to $X$ inside $Y$. These intersect along the projectivization of the normal cone, which appears in $Z\_1$ as the exceptional divisor, and in $Z\_2$ as the stuff added in projective completion.


(An example, for people who like polytopal pictures of toric varieties: let $Y$ be ${\mathbb P}^2$, pictured as a triangle, and $X$ be a point, pictured as a vertex. Then $Y \times {\mathbb P}^1$ is pictured as a triangular prism, and its blowup by cutting a corner off that prism. The fiber over $\infty$ is then pictured as cutting that triangle into a trapezoid $Z\_1$ union a small triangle $Z\_2$, glued along an interval, matching the decomposition above.)


So, if perhaps you don't like degenerating $Y$ (which may be complete) to something noncomplete, you can complete it by including $Z\_1$. The normal cone is $Z\_2 \setminus Z\_1$.


Two other comments. On $Y \times {\mathbb P}^1$ there is a circle action, dilating the ${\mathbb P}^1$. It acts trivially on the $0$ and $\infty$ fibers, moving the rest around. When we blow up $X \times \infty$, the circle action on the new $\infty$ fiber $Z\_1 \cup Z\_2$ is nontrivial on $Z\_2$; it is the dilation of the cone fibers. This is one place to lay blame for the existence of this "cone" structure.


Finally, there's a conformally equivalent way to think about the $Y$ to $Z\_1 \cup Z\_2$ degeneration, at least when $X$ and $Y$ are smooth. Pick a small tubular neighborhood (nonalgebraic!) around $X$, with boundary $S$, a sphere bundle over $X$. Let the metric on $Y$ get very looong nearby $S$, in directions passing through $S$. You might say that $X$ is falling into a black hole, with $S$ the Schwarzschild boundary. In the limit, it gets infinitely long, and $X$ has bubbled off into its own universe. 


