Number of cuts (algebraic geometry)

In algebraic geometry , the intersection number denotes a positive integer, which denotes the multiplicity of intersections of intersection points of algebraic curves.

definition

• Let be an algebraically closed field and let and be plane affine algebraic curves in . The number of cuts from and in the point is denoted by and is defined by: ${\displaystyle k}$ ${\displaystyle F}$ ${\displaystyle G}$ ${\displaystyle k^{2}}$ ${\displaystyle F}$ ${\displaystyle G}$ ${\displaystyle P\in k^{2}}$ ${\displaystyle I(P,F\cap G)}$

${\displaystyle I(P,F\cap G):=\dim _{k}\left({\mathcal {O}}_{P}(k^{2})/(F,G)\right)}$

Here denotes the ring of regular functions of the affine variety located at the point . ${\displaystyle {\mathcal {O}}_{P}(k^{2})}$ ${\displaystyle P}$ ${\displaystyle {\mathcal {O}}(k^{2})}$ ${\displaystyle k^{2}}$

• ${\displaystyle F}$and actually cut themselves in when they have no common component that contains. ${\displaystyle G}$ ${\displaystyle P}$ ${\displaystyle P}$
• ${\displaystyle F}$and intersect transversely in when a single point is both curves and the tangents of the two curves are different in this point. ${\displaystyle G}$ ${\displaystyle P}$ ${\displaystyle P}$

properties

The number of cuts has the following properties:

1. If and in actually intersect is a non-negative integer, otherwise is . ${\displaystyle F}$ ${\displaystyle G}$ ${\displaystyle P}$ ${\displaystyle I(P,F\cap G)}$ ${\displaystyle I(P,F\cap G)=\infty }$
2. ${\displaystyle I(P,F\cap G)=0\Longleftrightarrow P\not \in F\cap G}$and is only dependent on the components of and which go through . ${\displaystyle I(P,F\cap G)}$ ${\displaystyle F}$ ${\displaystyle G}$ ${\displaystyle P}$
3. Let be an affine coordinate transformation of with , then: ${\displaystyle T}$ ${\displaystyle k^{2}}$ ${\displaystyle T(Q)=P}$ ${\displaystyle I(P,F\cap G)=I(Q,F\circ T\cap G\circ T)}$
4. ${\displaystyle I(P,G\cap F)=I(P,F\cap G)}$
5. ${\displaystyle I(P,F\cap G)\geq m_{P}(F)\cdot m_{P}(G)}$with equality if and only if and in have no common tangents. ${\displaystyle F}$ ${\displaystyle G}$ ${\displaystyle P}$
6. If and , then: ${\displaystyle F=\prod _{i=1}^{n}F_{i}^{r_{i}}}$ ${\displaystyle G=\prod _{i=1}^{m}G_{j}^{s_{j}}}$ ${\displaystyle I(P,F\cap G)=\sum _{i,j}r_{i}s_{j}I(P,F_{i}\cap G_{j})}$
7. ${\displaystyle I(P,F\cap G)=I(P,F\cap (G+AF))\quad \forall A\in {\mathcal {O}}(k^{2})}$
8. If is a single point of , then holds . ${\displaystyle P}$ ${\displaystyle F}$ ${\displaystyle I(P,F\cap G)=\mathrm {ord} _{P}^{F}(G)}$
9. When and no common components have, then: ${\displaystyle F}$ ${\displaystyle G}$ ${\displaystyle \sum _{P\in k^{2}}I(P,F\cap G)=\dim _{k}\left({\mathcal {O}}(k^{2})/(F,G)\right)}$

The number of cuts is also clearly determined by these properties.

example

Let be an algebraically closed field of characteristic and as well . The following intersections are found: ${\displaystyle k}$ ${\displaystyle 0}$ ${\displaystyle F=(X-1)(Y^{2}-X^{3})^{2}}$ ${\displaystyle G=(X-1)^{2}(Y^{2}-X^{3}-X^{2})}$

• ${\displaystyle (1,y),\;y\in k}$. In this case the points lie in a common component of and , so: ${\displaystyle X-1}$ ${\displaystyle F}$ ${\displaystyle G}$ ${\displaystyle I((1,y),F\cap G)=\infty }$
• ${\displaystyle (0,0)}$: Using the properties of the number of cuts, one calculates:

${\displaystyle I((0,0),F\cap G)=I((0,0),(Y^{2}-X^{3})^{2}\cap (Y^{2}-X^{3}-X^{2}))}$

${\displaystyle =2\cdot I((0,0),(Y^{2}-X^{3})\cap (Y^{2}-X^{3}-X^{2}))=2\cdot I((0,0),(Y^{2}-X^{3})\cap X^{2})}$

${\displaystyle =2\cdot m_{(0,0)}(Y^{2}-X^{3})\cdot m_{(0,0)}(-X^{2})=2\cdot 2\cdot 2=8}$

Bézout's theorem

By introducing homogeneous coordinates, the definition of the number of cuts can be extended to projective plane curves. The set of Bézout then states that for projective plane curves applicable without common components: ${\displaystyle F,G}$

${\displaystyle \sum _{P\in \mathbb {P} ^{2}(k)}I(P,F\cap G)=\deg F\cdot \deg G}$

If one restricts oneself to affine plane curves without common components, only the inequality applies:

${\displaystyle \sum _{P\in k^{2}}I(P,F\cap G)\leq \deg F\cdot \deg G}$

generalization

A generalization to varieties of higher dimensions is possible, see the work "Intersection Theory" by William Fulton , which was awarded the Leroy P. Steele Prize .

See also

• Number of cuts

literature

• William Fulton: Algebraic Curves. An Introduction to Algebraic Geometry. Mathematics lecture note series, 30. Benjamin / Cummings, New York 1969, ISBN 0-201-51010-3
• William Fulton: Intersection Theory. Results of mathematics and its border areas. Episode 3. Springer, Berlin 1998, ISBN 3-540-62046-X