 Right, so welcome back to the second part of Kirsten talk. Thank you. So, let's go back to. We have these non intersecting lines one through L four in P three and let's go back to counting L intersecting lines in P three. So, we have whoops, what happened? There, sorry. Okay. We've got an equation for each line. That is a section of the dual tautological line. The dual tautological bundle wedge itself over the gross monion. This is Sigma and this actually breaks up Sigma one Sigma two Sigma three Sigma four. And we know that our lines that intersect are the zeros of this section. And we were just talking about our Euler number, which can be computed with our section with isolated zeros, which is the sum over these lines L in P three. Intersection actually intersecting the li of this local degree. And now we can do the same trick we did with the changing the Sigma. So if we change the section to the section where we took minus Sigma one Sigma two Sigma three Sigma four we can all of our Jacobians change sign. So it multiplies EV by minus one. And while we're at it, we can multiply EV by a which means we have a EV is EV and you can jack this up to a proof that EV has to be a multiple of the hyperbolic element which had the special property that when you multiplied by a you didn't change the hyperbolic element. So is a multiple of H and look comparing the rank comparing with the answer to over C. So this is an integer multiple. Okay, so is there a geometric interpretation of the of the local terms so question is there geometric interpretation the right hand side and the right hand side and what kind of geometric information is available we have intersection points L intersect gives four points on L which is a P one over its field of definition. So they have a cross ratio of L be defined to be the cross ratio of L I in the same order that we do the section sections and this is in K of L. We also have the planes spanned by L and Li so that P I be the plane and planes in P three containing L let's say L is the projectivization of a two-dimensional space W are three-dimensional subspaces V which are in between W and K of V to the fourth where P I is the projectivization of say V of such space of such planes is again a P one of the field of definition of the point corresponding to W which is also getting called L because they this is in correspondence with the four-dimensional space of the four-dimensional space by two-dimensional space so we get a two-dimensional space here so we get a one-dimensional space of planes containing L so the P I then have a cross ratio as well plus P one, P two, P three P four it's four points on a P one have a cross ratio P one, K of L and let's let U L be their cross ratio and we have this cross ratio and you can show that the local degree is the transfer of lambda L minus U L and putting this together we have the result of Padma and me that given four lines O one, L two, O three, L four P four general lines and P three, K or K is a field of characteristic not two and then we get a count with some arithmetic information of the lines L meeting O two, O three O four the trace of this difference of pork cross ratios and brackets has to be the hyperbolic element in G W of K we can if you take co-dimension two planes in P N you can get a similar result so what pi one pi two N minus two the co-dimension two planes P N or N odd then there's a result that says the lines meeting I I for all I of the trace or transfer of the lines L a messier quantity of I two N minus two matrix of dot products C I B one I B I B two I and this is equal to one over two N two N minus two N minus one times the hyperbolic element and these C I and B I are certain coordinates for the plane spanned by PI and L and B one I B two I are coordinates for L intersect and B one K of L and and as a corollary over F Q you cannot have a line defined over F Q squared meeting the four lines with the cross ratio lambda minus you equal to a non square or Q congruent to three mod four and for a square and with Q congruent to one mod four and there's and related results giving some sugar calculus for these folks due to Matthias and and so this is the theorem that I'm really going to have time to say but I want to call attention to the fact that these ones and minus ones are appearing because of the Z so we mentioned that the growth in the group of Z is actually isomorphic by pullback to that of our and one can show that there is a a series that was put on archive about permission K theory without inverting to so let X be a smooth and proper over Z join one over D factorial or some D could be one and let V be a relatively oriented vector bundle with the dimension the same as the rank so that our Euler number lives in the GW that that we were considering before so then the Euler number is in the adjoin minus one to primes up to D or numbers up to D and GW of K so in particular will for things to find over over Z the global counts have these minus ones and ones there are some cool enumerative setups where you get forms that absolutely cannot be expressed with the ones with the minus ones but I am out of time so I will stop here thank you so much thank you very much