Rational cuspidal plane curves of type (d,d-4) with

In the present paper we classify rational cuspidal plane curves with maximal multiplicity deg C - 4 and at least three cusps and where (V,D) is the minimal (SNC) resolution of (ℙ²,C). Received: 28 August 1998     

Counting curves on rational surfaces

In [CH3], Caporaso and Harris derive recursive formulas counting nodal plane curves of degree d and geometric genus g in the plane (through the appropriate number of fixed general points). We rephrase their arguments in the language of maps, and extend them to other rational surfaces, and other specified intersections with a divisor. As applications, (i) we count irreducible curves on Hirzebruch surfaces in a fixed divisor class and of fixed geometric genus, (ii) we compute the higher-genus Gromov–Witten invariants of (or equivalently, counting curves of any genus and divisor class on) del Pezzo surfaces of degree at least 3. In the case of the cubic surface in (ii), we first use a result of Graber to enumeratively interpret higher-genus Gromov–Witten invariants of certain K-nef surfaces, and then apply this to a degeneration of a cubic surface. Received: 30 June 1999 / Revised version: 1 January 2000     

Deformations of plane curves and Jacobian syzygies

We relate the equianalytic and the equisingular deformations of a reduced complex plane curve to the Jacobian syzygies of its defining equation. Several examples and conjectures involving rational cuspidal curves are discussed.     

Some general results on plane curves with total inflection points

In this paper we study plane curves of degree d with e total inflection points, for nonzero natural numbers d and e. Marc Coppens: the author is affiliated with K. U. Leuven as Research Fellow Received: 25 October 2006     

Curves in the double plane

We study in detail locally Cohen-Macaulay curves in P⁴ which are contained in a double plane 2H, thus completing the classification of curves lying on surfaces of degree two. We describe the irreducible components of the Hilbert schemes H _d,g(2H) of lo-cally Cohen-Macaulay curves in 2H of degree d and arithmetic genus g, and we show that H _d,g(2H) is connected. We also discuss the Rao module of these curves and liaison and biliaison equiva-lence classes.     

Rational Cuspidal Plane Curves of Type (d,d — 3)

In the previous paper [FlZa 2] we classified the rational cuspidal plane curves C with a cusp of multiplicity deg C — 2. In particular, we showed that any such curve can be transformed into a line by Cremona transformations. Here we do the same for rational cuspidal plane curves C with a cusp of multiplicity deg C — 3.     

Residual intersection theory with reducible schemes

We develop a formula (Theorem 5.2) which allows to compute top Chern classes of vector bundles on the vanishing locus V(s) of a section of this bundle. This formula particularly applies in the case when V(s) is the union of local complete intersections giving the individual contribution of each component and their mutual intersections. We conclude with applications to the enumeration of rational curves in complete intersections in projective space.     

On the number of cusps on cuspidal curves on Hirzebruch surfaces

In this article we give an upper bound for the number of cusps on a cuspidal curve on a Hirzebruch surface. We adapt the results that have been found for a similar question asked for cuspidal curves on the projective plane, and restate the results in this new setting.     

Plane curves whose tangent lines at collinear points are concurrent

We are interested in a particular geometry of plane curves in characteristicp>0, which was inspired by Thas's article [13]. We will prove that any plane curve of degree > 2 whose tangent lines at collinear points are concurrent is either a strange curve or projectively equivalent to the Fermat curve of degreeq + 1, whereq is a power ofp.     

Remarks on families of singular curves with hyperelliptic normalizations

We give restrictions on the existence of families of curves on smooth projective surfaces S of nonnegative Kodaira dimension all having constant geometric genus p_g ? 2 and hyperelliptic normalizations. In particular, we prove a Reider-like result that relies on deformation theory and bending-and-breaking of rational curves in Sym²(S). We also give examples of families of such curves.     

Kontsevich's formula and the WDVV equations in tropical geometry

Using Gromov-Witten theory the numbers of complex plane rational curves of degree d through 3d−1 general given points can be computed recursively with Kontsevich's formula that follows from the so-called WDVV equations. In this paper we establish the same results entirely in the language of tropical geometry. In particular this shows how the concepts of moduli spaces of stable curves and maps, (evaluation and forgetful) morphisms, intersection multiplicities and their invariance under deformations can be carried over to the tropical world.     

Relative Gromov-Witten invariants and the mirror formula

Let X be a smooth complex projective variety, and let be a smooth very ample hypersurface such that is nef. Using the technique of relative Gromov-Witten invariants, we give a new short and geometric proof of (a version of) the “mirror formula”, i.e. we show that the generating function of the genus zero 1-point Gromov-Witten invariants of Y can be obtained from that of X by a certain change of variables (the so-called “mirror transformation”). Moreover, we use the same techniques to give a similar expression for the (virtual) numbers of degree-d plane rational curves meeting a smooth cubic at one point with multiplicity 3d, which play a role in local mirror symmetry. Received: 11 July 2001 / Published online: 4 February 2003 Funded by the DFG scholarships Ga 636/1–1 and Ga 636/1–2.     

Naturality of Abel maps

We give a combinatorial characterization of nodal curves admitting a natural (i.e. compatible with and independent of specialization) dth Abel map for any d ≥ 1.     

On Jacquet-Langlands isogeny over function fields

We propose a conjectural explicit isogeny from the Jacobians of hyperelliptic Drinfeld modular curves to the Jacobians of hyperelliptic modular curves of D-elliptic sheaves. The kernel of the isogeny is a subgroup of the cuspidal divisor group constructed by examining the canonical maps from the cuspidal divisor group into the component groups.     

Hyperellipticity of offsets to rational plane curves

In this paper, we prove irreducible offsets to rational plane curves are hyperelliptic in general and compute the genus of them. We also give a criterion for deciding the irreducibility of offsets to rational plane curves.     

Towards the cohomology of moduli spaces of higher genus stable maps

We prove that the orbifold desingularization of the moduli space of stable maps of genus g = 1 recently constructed by Vakil and Zinger has vanishing rational cohomology groups in odd degree k < 11. Received: 29 January 2007     

Rational curves of degree 16 on a general heptic fourfold

According to a conjecture of H. Clemens, the dimension of the space of rational curves on a general projective hypersurface should equal the number predicted by a naïve dimension count. In the case of a general hypersurface of degree 7 in P⁵${\mathbb{P}}^5$, the conjecture predicts that the only rational curves should be lines. This has been verified by Hana and Johnsen for rational curves of degree at most 15. Here we extend their results to show that no rational curves of degree 16 lie on a general heptic fourfold.