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We report on the problem of the existence of complex and real algebraic curves in the plane with prescribed singularities up to analytic and topological equivalence. The question is whether, for a given positive integer $d$ and a finite number of given analytic or topological singularity types, there exist a plane (irreducible) curve of degree $d$ having singular points of the given type as its only singularities. The set of all such curves is a quasi-projective variety, which we call an equisingular family (ESF). We describe, in terms of numerical invariants of the curves and their singularities, the state of the art concerning necessary and sufficient conditions for the non-emptiness and $T$-smoothness (i.e., smooth of expected dimension) of the corresponding ESF. The considered singularities can be arbitrary, but we spend special attention to plane curves with nodes and cusps, the most studied case, where still no complete answer is known in general. An important result is, however, that the necessary and the sufficient conditions show the same asymptotics for $T$-smooth equisingular families if the degree goes to infinity.
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Let G be a finite group, and $g geq 2$. We study the locus of genus g curves that admit a G-action of given type, and inclusions between such loci. We use this to study the locus of genus g curves with prescribed automorphism group G. We completely classify these loci for g=3 (including equations for the corresponding curves), and for $g leq 10$ we classify those loci corresponding to large G.
This paper discusses a central theorem in birational geometry first proved by Eugenio Bertini in 1891. J.L. Coolidge described the main ideas behind Bertinis proof, but he attributed the theorem to Clebsch. He did so owing to a short note that Felix Klein appended to the republication of Bertinis article in 1894. The precise circumstances that led to Kleins intervention can be easily reconstructed from letters Klein exchanged with Max Noether, who was then completing work on the lengthy report he and Alexander Brill published on the history of algebraic functions [Brill/Noether 1894]. This correspondence sheds new light on Noethers deep concerns about the importance of this report in substantiating his own priority rights and larger intellectual legacy.
We provide a lower bound on the degree of curves of the projective plane $mathbb{P}^2$ passing through the centers of a divisorial valuation $ u$ of $mathbb{P}^2$ with prescribed multiplicities, and an upper bound for the Seshadri-type constant of $ u$, $hat{mu}( u)$, constant that is crucial in the Nagata-type valuative conjecture. We also give some results related to the bounded negativity conjecture concerning those rational surfaces having the projective plane as a relatively minimal model.
This note pursues the techniques of modified psi classes on the stack of stable maps (cf. [Graber-Kock-Pandharipande]) to give concise solutions to the characteristic number problem of rational curves in P^2 or P^1 x P^1 with a cusp or a prescribed triple contact. The classes of such loci are computed in terms of modified psi classes, diagonal classes, and certain codimension-2 boundary classes. Via topological recursions the generating functions for the numbers can then be expressed in terms of the usual characteristic number potentials.
The local invariants of a meromorphic Abelian differential on a Riemann surface of genus $g$ are the orders of zeros and poles, and the residues at the poles. The main result of this paper is that with few exceptions, every pattern of orders and residues can be obtain by an Abelian differential. These exceptions are two families in genus zero when the orders of the poles are either all simple or all nonsimple. Moreover, we even show that the pattern can be realized in each connected component of strata. Finally we give consequences of these results in algebraic and flat geometry. The main ingredient of the proof is the flat representation of the Abelian differentials.
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