Cayley-Bacharach theorem

Two cubes (red and blue, here specifically three straight lines each) intersect at nine points. Every additional cubic (black) that goes through eight of these nine points already contains the ninth point.

The set of Cayley Bacharach is a mathematical theorem in the area of the algebraic geometry . It makes a statement that in certain cases algebraic curves that go through some of the intersection points of two further algebraic curves already contain all of these intersection points. In particular, a cubic curve that goes through eight of nine intersections of two other cubes also contains the last intersection. This statement was formulated and proven for the first time by Michel Chasles , the sentence is usually named after Arthur Cayley and Isaak Bacharach , who proposed or proved generalizations of the statement.

statement

In Chasles' formulation, the sentence says the following:

If two cubic curves intersect in the projective plane at nine different points, every cubic curve that passes through eight of these points also contains the ninth.

According to Bézout's theorem , 9 is the maximum possible number of different intersection points, provided that the two curves do not have a common component. This maximum number is always reached over an algebraically closed field if the points are all different.

Cayley made a generalization of the theorem. In the original version, however, important conditions are missing, and his proof also contained several loopholes. Building on the work of Alexander von Brill and Max Noether, Bacharach was able to remedy these deficiencies and presented a correct generalization in his inaugural lecture in 1881. In a later publication he formulated the generalization as follows:

Intersect two algebraic curves of the orders and in different points, each algebraic curve as containing of the order of , and , up through all the on going of these points, these remaining points; unless these points lie on a curve of order . ${\displaystyle d_{1}}$ ${\displaystyle d_{2}}$ ${\displaystyle d_{1}d_{2}}$ ${\displaystyle k}$ ${\displaystyle k\geq d_{1}}$ ${\displaystyle k\geq d_{2}}$ ${\displaystyle k\leq d_{1}+d_{2}-3}$ ${\displaystyle {d_{1}+d_{2}-k-1 \choose 2}}$ ${\displaystyle {d_{1}+d_{2}-k-1 \choose 2}}$ ${\displaystyle d_{1}+d_{2}-k-3}$

For Chasles' sentence results. ${\displaystyle d_{1}=d_{2}=k=3}$

Proof idea

If a set of points is in the projective plane, the polynomials of a certain degree , which vanish at all points of , form a vector space . The codimension of this vector space in the vector space of all polynomials of degree indicates how much the choice of an algebraic curve of degree restricts through the points. ${\displaystyle \Gamma }$ ${\displaystyle d}$ ${\displaystyle \Gamma }$ ${\displaystyle d}$ ${\displaystyle \Gamma }$ ${\displaystyle d}$

For points in general position one expects that this codimension corresponds to the number of points, because each point places a linear condition on the polynomial.

The vector space of all homogeneous polynomials in three variables of degree has the dimension , in the case of cubics so dimension 10. Refers to the amount of the nine intersections and an 8-element subset, so therefore you would expect for a codimension of 8. But also for results a codimension of a maximum of 8, since with the two polynomials that define the two given cubics, there are already two linearly independent polynomials that vanish at all points of . ${\displaystyle d}$ ${\displaystyle {d+2 \choose 2}}$ ${\displaystyle \Gamma }$ ${\displaystyle \Gamma '\subset \Gamma }$ ${\displaystyle \Gamma '}$ ${\displaystyle \Gamma }$ ${\displaystyle \Gamma }$

In fact, one can show that the codimension for and coincide and that every cube through the points of already goes through all the points of . ${\displaystyle \Gamma }$ ${\displaystyle \Gamma '}$ ${\displaystyle \Gamma '}$ ${\displaystyle \Gamma }$

Applications

Theorems of Pappos and Pascal

Pascal's theorem

Both Pappos' and Pascal's theorem are special cases of Cayley-Bacharach's theorem. If there are six points on a conic section , the three straight lines , and, on the one hand, and , and, on the other hand, form two cubics that intersect at nine points, namely in and in the three intersection points , and . The conic section, together with the line through and also a cube, it passes through eight of the points, after the set of Cayley-Bacharach so by . So , and are collinear, this is Pascal's theorem. The Pappos theorem can be derived analogously. ${\displaystyle P_{1},\dots ,P_{6}}$ ${\displaystyle P_{1}P_{2}}$ ${\displaystyle P_{3}P_{4}}$ ${\displaystyle P_{5}P_{6}}$ ${\displaystyle P_{2}P_{3}}$ ${\displaystyle P_{4}P_{5}}$ ${\displaystyle P_{6}P_{1}}$ ${\displaystyle P_{1},\dots ,P_{6}}$ ${\displaystyle P_{7}}$ ${\displaystyle P_{8}}$ ${\displaystyle P_{9}}$ ${\displaystyle P_{7}}$ ${\displaystyle P_{8}}$ ${\displaystyle P_{9}}$ ${\displaystyle P_{7}}$ ${\displaystyle P_{8}}$ ${\displaystyle P_{9}}$

Group operation on elliptic curves

Addition of points on an elliptic curve

With the help of Cayley-Bacharach's theorem, it is easy to prove the associative law for addition on elliptic curves : Let , and three points on an elliptic curve, be the point that represents the neutral element . Then the three straight lines , and form a cube, likewise the three straight lines , and . The points of intersection of these two cubics are , , , , , , (on the straight and ) (on the straight and ), and the intersection of and . The elliptical curve contains the first eight points, including the last. This must therefore be what applies. ${\displaystyle P}$ ${\displaystyle Q}$ ${\displaystyle R}$ ${\displaystyle O}$ ${\displaystyle QR}$ ${\displaystyle O(P+Q)}$ ${\displaystyle P(Q+R)}$ ${\displaystyle PQ}$ ${\displaystyle O(Q+R)}$ ${\displaystyle R(P+Q)}$ ${\displaystyle P}$ ${\displaystyle Q}$ ${\displaystyle R}$ ${\displaystyle O}$ ${\displaystyle P+Q}$ ${\displaystyle Q+R}$ ${\displaystyle -P-Q}$ ${\displaystyle O(P+Q)}$ ${\displaystyle PQ}$ ${\displaystyle -Q-R}$ ${\displaystyle QR}$ ${\displaystyle O(Q+R)}$ ${\displaystyle P(Q+R)}$ ${\displaystyle R(P+Q)}$ ${\displaystyle -P-(Q+R)=-R-(P+Q)}$ ${\displaystyle (P+Q)+R=P+(Q+R)}$

Individual evidence

1. ↑ Michel Chasles: Traité des sections coniques. Gauthier-Villars, Paris, 1865. ( digitized version )
2. ^ Arthur Cayley: On the Intersection of Curves. In: Cambridge Mathematical Journal. Volume 3, 1843. pp. 211-213. ( Digitized version )
3. ^ David Eisenbud , Mark Green , Joe Harris : Cayley-Bacharach Theorems and Conjectures. In: Bulletin of the American Mathematical Society. Volume 33, No. 3, July 1996. ( online , PDF)
4. ↑ Isaak Bacharach: About Cayley's intersection theorem. In: Mathematical Annals . Volume 26, 1886. pp. 275-299. ( doi: 10.1007 / BF01444338 )

Web links

• Eric W. Weisstein : Cayley-Bacharach Theorem . In: MathWorld (English).