Enumeration of singular algebraic curves
نویسنده
چکیده
We enumerate plane algebraic curves with one singular point of any (prescribed) singularity type. It is shown how to generalize the method to the singular hypersurfaces and some cases of enumeration of singular hypersurfaces are solved.
منابع مشابه
A Tropical Approach to Enumerative Geometry
A detailed algebraic-geometric background is presented for the tropical approach to enumeration of singular curves on toric surfaces, which consists of reducing the enumeration of algebraic curves to that of non-Archimedean amoebas, the images of algebraic curves by a real-valued non-Archimedean valuation. This idea was proposed by Kontsevich and recently realized by Mikhalkin, who enumerated t...
متن کاملPatchworking singular algebraic curves, non-Archimedean amoebas and enumerative geometry
We prove new patchworking theorems for singular algebraic curves, which state the following. Given a complex toric threefold Y which is fibred over C with a reduced reducible zero fiber Y0 and other fibers Yt smooth, and given a curve C0 ⊂ Y0, the theorems provide sufficient conditions for the existence of one-parametric family of curves Ct ⊂ Yt, which induces an equisingular deformation for so...
متن کاملEnumerating singular curves on surfaces
We enumerate the singular algebraic curves in a complete linear system on a smooth projective surface. The system must be suitably ample in a rather precise sense. The curves may have up to eight nodes, or a triple point of a given type and up to three nodes. The curves must also pass through appropriately many general points. The number of curves is given by a universal polynomial in four basi...
متن کاملEnumeration of uni-singular algebraic hypersurfaces
We enumerate complex algebraic hypersurfaces in Pn, of a given (high) degree with one singular point of a given singularity type. Our approach is to compute the (co)homology classes of the corresponding equi-singular strata in the parameter space of hypersurfaces. We suggest an inductive procedure, based on intersection theory combined with liftings and degenerations. The procedure computes the...
متن کاملOn the enumeration of complex plane curves with two singular points
We study equi-singular strata of curves with two singular points of prescribed types. The method of our previous work [Kerner04] is generalized to this case. This allows to solve the enumerative problem for plane curves with two singular points of linear singularity types. In the general case this reduces the enumerative questions to the problem of collision of the two singular points. The meth...
متن کامل