Hebrew University topology and geometry seminar

Abstract: Mikhalkin changed the way of considering enumerative problems in algebraic geometry when he considered tropical curves counted with a numerical multiplicity and proved that the invariance of classical enumerative numbers can be proven tropically. Block and Göttsche recently introduced polynomial multiplicities (in the variable y) for plane tropical curves which yield an invariant number when we count curves passing through a generic point configuration. In addition, they reveal deep relations to classical enumerative problems: specializing y=1 we obtain the corresponding Gromov Witten invariant and for y=-1 we retrieve the corresponding Welschinger invariant. In this talk I present a similar approach for broccoli invariants which have been introduced to prove the invariance of tropical Welschinger numbers for certain real curves. Endowing each broccoli curve with a polynomial multiplicity yields again an invariant and this approach can be used to find simpler Caporaso Harris type formulas for broccoli invariants. This is joint work in progress with Lothar Goettsche.