3:45PM: Complete twisted cubics
Francesco CavazzaniHow many plane conics are tangent to 5 general conics? How many twisted cubics in the projective space are tangent to 12 general planes? Enumerative problems like these go all the way back to the middle of 19th century. Many times in history, the answer came finding the right parameter space for the objects in question; the answer to the first question (3264) was in fact solved considering the space of complete conics. For twisted cubics, on the other hand, there is no such space of “complete twisted cubics”. The moduli spaces that are currently most used, the Hilbert scheme and the Kontsevich space, are missing many of the properties that makes the space of complete conics, in some sense, wonderful. In this talk, I will show how to obtain a different and more ``symmetric’’ moduli space for twisted cubics; here, with the help of some representation theory, it is possible to give answers to some enumerative questions like the above.