Degree of intersection curve of two quadrics - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T11:35:34Z http://mathoverflow.net/feeds/question/100207 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/100207/degree-of-intersection-curve-of-two-quadrics Degree of intersection curve of two quadrics Tim Southee 2012-06-21T05:50:18Z 2012-06-22T14:16:17Z <p>Given two quadrics in $RP^3$ of the following form</p> <p>$q_1(Z_1^2+Z_2^2)+q_2(Z_1Z_3-Z_2Z_4)+q_3(Z_2Z_3+Z_1Z_4) +q_4(Z_1Z_3+Z_2Z_4)$ $+ q_5(Z_2Z_3-Z_1Z_4)+q_6Z_3Z_4 +q_7(Z_3^2-Z_4^2)+q_{8}(Z_3^2+Z_4^2)=0,$</p> <p>where coefficients $q_1=0, q_2 = 2a_0l_1, q_3 = 2a_0l_2, q_4 = 0, q_5 = 0, q_6 = -2a_1l_2+2a_2l_1, q_7=-a_1l_1-a_2l_2, q_8 = a_0l_3$, and $a_i, l_i$ are known for the two quadrics, how do I find the degree of the intersection curve of the two quadrics? Bezout theorem tells us that the maximum degree would be 4, but in this case, I am told that it is 3. How?</p> <p>A more general question would be: is there an analytical method to find the degree of an intersection curve of any two given quadrics not necessarily in the above form?</p> http://mathoverflow.net/questions/100207/degree-of-intersection-curve-of-two-quadrics/100358#100358 Answer by Jérémy Blanc for Degree of intersection curve of two quadrics Jérémy Blanc 2012-06-22T14:16:17Z 2012-06-22T14:16:17Z <p>Let us describe the general case first, before going to your special case. The intersection of two quadrics has always degree $4$, if you count the intersection with multiplicities. So if you want something of degree $3$, it means that it is a conic (degree $2$) and a line counted with multiplicity $2$, or three lines where one is counted with multiplicity $2$.</p> <p>If you want to find how is the intersection, you can choose one quadric to be smooth, and then isomorphic to $\mathbb{P}^1\times \mathbb{P}^1$ over $\mathbb{C}$, and restricts the equation of the second one to the first one. You get an equation of bidegree $2,2$ and check if it decomposes. If the quadric is a cone you can do a similar argument.</p> <p>But in general, if you have the explicit equation it is not hard to directly check if some line is contained in both quadrics.</p> <p>Let us go back to your equation. It is quite strange written, since you let coefficients which are zero, but it corresponds simply to two equation of the form</p> <p>$q_2(Z_1Z_3-Z_2Z_4)+q_3(Z_2Z_3+Z_1Z_4)+q_6Z_3Z_4+q_7(Z_3^2-Z_4^2)+q_8(Z_3^2+Z_4^2)$</p> <p>Your other assumptions on the $q_i$'s do not seem to be really special. If you have no conditions on the $a_i,l_i$, the $q_i$ above can be anything.</p> <p>As Johannes observed, the line $Z_3=Z_4=0$ is contained in your quadrics. This implies that the intersection contains the line (counted maybe with multiplicity) and the remains curve has degree $3$. If you check correctly you will see that there is no reason that in fact the intersection has degree $3$ in your case; it is easy to find coefficients such that the intersection is the line $Z_3=Z_4$ and an irreducible cubic.</p> <p>If someone told you that the intersection had degree $3$, he probably meant that the intersection is the line AND a curve of degree $3$.</p>