(Degree)
Let \(X \subseteq \PP^N\) be projective of dimension \(n\). The degree of \(X\) is the number of points of \[ X \intersect H_1 \intersect \cdots \intersect H_n \] for \(H_1, \ldots, H_n\) general hyperplanes.
This number is finite, independent of the general hyperplanes chosen, and equals \(n!\) times the leading coefficient of the Hilbert polynomial \(P_X\).
If a hyperplane \(H\) contains no irreducible component of \(X\), the sequence \(0\to\OO_X(-1)\to\OO_X\to\OO_{X\cap H}\to0\) is exact, so \(P_{X\cap H}(r)=P_X(r)-P_X(r-1)\). If \(P_X\) has leading term \(\frac{d}{n!}r^n\), then \(P_{X\cap H}\) has leading term \(\frac{d}{(n-1)!}r^{n-1}\), so \(X\cap H\) has dimension \(n-1\) and the same degree \(d\). After \(n\) such cuts, \(X\cap H_1\cap\cdots\cap H_n\) is a finite scheme of length \(d\); for general hyperplanes it is reduced, so it consists of \(d\) points.
For hyperplanes that are not general, the intersection can have fewer points: a line tangent to a smooth conic meets it in one point, of intersection multiplicity \(2\), as Bézout’s theorem requires. Degree depends on the embedding: the twisted cubic and a line are both isomorphic to \(\PP^1\) but have degrees \(3\) and \(1\) in their respective embeddings.
Dependencies
D-L6ERWThe Hilbert polynomial and its coefficientsuses
Backlinks
D-CRVDEGREESDegrees of points, finite morphisms, line bundles and coherent sheaves on curvesrelated-toD-VARCHOWChow forms and the Chow varietyusesD-SRFINTThe intersection pairing on a surfaceuses