01201nas a2200145   4500008004100000245005000041210005000091260003400141300001400175490000700189520075600196100002200952700003100974856005001005       2016                        eng d00aRefined node polynomials via long edge graphs0 aRefined node polynomials via long edge graphs  bInternational Press of Boston  a193–2340 v103 a<p>The generating functions of the Severi degrees for sufficiently ample line bundles on algebraic surfaces are multiplicative in the topological invariants of the surface and the line bundle. Recently new proofs of this fact were given for toric surfaces by Block, Colley, Kennedy and Liu, Osserman, using tropical geometry and in particular the combinatorial tool of long-edged graphs. In the first part of this paper these results are for $\mathbb{P}^2$ and rational ruled surfaces generalised to refined Severi degrees. In the second part of the paper we give a number of mostly conjectural generalisations of this result to singular surfaces, and curves with prescribed multiple points. The formulas involve modular forms and theta functions.</p>1 aGöttsche, Lothar1 aKikwai, Benjamin, Kipkirui  uhttp://dx.doi.org/10.4310/CNTP.2016.v10.n2.a201121nas a2200121   4500008004300000245007400043210006900117260003700186520069600223100002200919700002200941856003600963       2010                        en_Ud  00aRiemann-Roch theorems and elliptic genus for virtually smooth schemes0 aRiemannRoch theorems and elliptic genus for virtually smooth sch  bMathematical Sciences Publishers3 aFor a proper scheme X with a fixed 1-perfect obstruction theory, we define virtual versions of holomorphic Euler characteristic, chi y-genus, and elliptic genus; they are deformation invariant, and extend the usual definition in the smooth case. We prove virtual versions of the Grothendieck-Riemann-Roch and Hirzebruch-Riemann-Roch theorems. We show that the virtual chi y-genus is a polynomial, and use this to define a virtual topological Euler characteristic. We prove that the virtual elliptic genus satisfies a Jacobi modularity property; we state and prove a localization theorem in the toric equivariant case. We show how some of our results apply to moduli spaces of stable sheaves.1 aFantechi, Barbara1 aGöttsche, Lothar  uhttp://hdl.handle.net/1963/3888