The Di Francesco-Itzykson-Göttsche Conjectures for Node Polynomials of $\mathbb{P}^{2}$

Details The Di Francesco-Itzykson-Göttsche Conjectures for Node Polynomials of $\mathbb{P}^{2}$ The Di Francesco-Itzykson-Göttsche Conjectures for Node Polynomials of $\mathbb{P}^{2}$

2010-10-12

arxiv.org/abs/1010.2377v4

Mathematics

Algebraic Geometry

17 pages. Updated to match the version to appear in Int. Journ. Math

Scientific paper

10.1142/S0129167X12500498

For a smooth, irreducible projective surface S over \mathbb{C}, the number of r-nodal curves in an ample linear system |L| (where L is a line bundle on S) can be expressed using the rth Bell polynomial P_{r} in r universal functions a_{i} of (S,L), which are linear polynomials in the four Chern numbers of S and L. We use this result to establish a proof of the classical shape conjectures of Di Francesco-Itzykson and G\"ottsche governing node polynomials in the case of P^{2}. We also give a recursive procedure which provides the L^{2}-term of the polynomials a_{i}.

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