Volume 9, Issue 2

In this paper, we consider weighted counts of tropical plane curves of particular combinatorial type through a certain number of generic points. We give a criterion, effectively balancing , derived from tropical intersection theory on the secondary fan, for a weighted count to give a number invariant of the position of the points. By computing a certain intersection multiplicity, we determine which weighted counts in our approach replicates Mikhalkin's computation of Gromov–Witten invariants although we do not know if such a count is effectively balanced. This begins to address a question raised by Dickenstein, Feichtner, and Sturmfels. We also give a geometric interpretation of the numbers we produce involving Chow quotients, and provide a counterexample showing that the tropical Severi variety is not always supported on the secondary fan.

In this paper we prove results on topological properties of quotients of manifolds, in particular of projective spaces and spheres. Specifically, under suitable conditions we exhibit results on homology and homotopy groups of these quotients. As an application we generalize the characterization of topological translation planes obtained by Löwen, Steinke, and Van Maldeghem in [Adv. Geom.: S59–S74, 2003] to all possible (finite) dimensions. Our approach also yields a more unified proof for this characterization result. Most of the results in this paper are obtained by means of standard results from algebraic topology.

We study two boundary value problems for a surface of revolution moving under Gauss curvature flow. The rotational symmetry allows us to reduce to an equation on the generating curve so that there is no restriction on the sign of the curvature of the initial surface.

We show that for a hypersurface Batyrev's stringy E -function can be seen as a residue of the Hodge zeta function, a specialization of the motivic zeta function of Denef and Loeser. This is a nice application of inversion of adjunction. If an affine hypersurface is given by a polynomial that is non-degenerate with respect to its Newton polyhedron, then the motivic zeta function and thus the stringy E -function can be computed from this Newton polyhedron (by work of Artal, Cassou-Noguès, Luengo and Melle based on an algorithm of Denef and Hoornaert). We use this procedure to obtain an easy way to compute the contribution of a Brieskorn singularity to the stringy E -function. As a corollary, we prove that stringy Hodge numbers of varieties with a certain class of strictly canonical Brieskorn singularities are nonnegative. We conclude by computing an interesting 6-dimensional example. It shows that a result, implying nonnegativity of stringy Hodge numbers in lower dimensional cases, obtained in our previous paper, is not true in higher dimension.

In [Dürr, Kabanov, Okonek, Topology 46: 225–294, 2007] we constructed virtual fundamental classes for Hilbert schemes of divisors of topological type m on a surface V , and used these classes to define the Poincaré invariant of V : We conjecture that this invariant coincides with the full Seiberg–Witten invariant computed with respect to the canonical orientation data. In this note we prove that the existence of an integral curve C ⊂ V induces relations between some of these virtual fundamental classes . The corresponding relations for the Poincaré invariant can be considered as algebraic analoga of the fundamental relations obtained in [Ozsváth, Szabó, Ann. of Math. 151: 93–124, 2000].

A smooth, projective surface S is said to be isogenous to a product if there exist two smooth curves C, F and a finite group G acting freely on C × F so that S = ( C × F )/ G . In this paper we classify all surfaces with p g = q = 1 which are isogenous to a product.

We construct a family of examples of Legendrian subvarieties in some projective spaces. Although most of them are singular, a new example of smooth Legendrian varieties in dimension 8 is in this family. This 8-fold has interesting properties: it is a compactification of the special linear group, a Fano manifold of index 5 and Picard number 1.

Given a smooth and oriented surface M in the Euclidean space ℝ 3 , the conjugate curve congruence C α is a family of pairs of foliations on M that links the lines of curvature and the asymptotic curves of M . This family is first introduced in [Fletcher, Geometrical problems in computer vision, Liverpool University, 1996] and is studied in [Bruce, Fletcher, Tari, Contemp. Math. 354: 1–18, 2004, Bruce, Tari, Trans. Amer. Math. Soc. 357: 267–285, 2005]. When the surface M = M 0 is deformed in a 1-parameter family of surfaces M t , we obtain a 2-parameter family of conjugate curve congruence C α,t . We study in this paper the generic local singularities in C α 0 ,0 and the way they bifurcate in the family C α,t , with ( α , t ) close to ( α 0 , 0).