We prove that the irreducible components of primitive class Severi varieties of general abelian surfaces are completely determined by the maximal factorization through an isogeny of the maps from the normalized curves. Contents 1. Introduction 1 2. The various moduli and parameter spaces 4 3. Partitions of rank two lattices 7 4. Maps from genus 2 curves to a genus 1 curve 8 5. The specialization to a split abelian surface 13 6. The degeneration of the moduli spaces 23 References 32

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We give an inductive proof that the generalized Severi varieties -the varieties which parametrize (irreducible) plane curves of given degree and genus, with a fixed tangency profile to a given line at several general fixed points and several mobile points -are irreducible.the situation is the complete opposite. On one hand, there have been only a few further developments concerning the irreducibility problem [14,15]. On the other hand, there has been a great amount of work and progress on the enumerative side, partly due to the availability of powerful tools such as (but not limited to) Li's degeneration formula [11], which allows one to systematically compute Caporaso-Harris type formulas.Given that the genesis of the generalized Severi varieties is quite intimately linked with the irreducibility problem, it is natural to ask whether they are irreducible. The answer is obviously no and the reason is that the curves are allowed to be reducible. However, this requirement (or lack thereof) is mostly a cosmetic one meant to simplify the combinatorics in [2]. It turns out that if we instead insist in the definition that the curves are irreducible (Definition 1.1), then the generalized Severi varieties are indeed irreducible, as expected. To the best of the author's knowledge, this statement does not appear anywhere in the literature, although it seems to lie within the realm of what is provable by adapting the arguments in the famous proof [8] by Harris discussed above. The objective purpose of the paper is to prove this statement (Theorem 1.6). However, we will not prove this statement by adapting the existing methods, so the subjective purpose is to outline and test a different approach to the irreducibility problem.Roughly, instead of using the known degenerations methods to prove a suitable version of step (3) above, we will use very similar methods to prove irreducibility directly. The long-term motivation is illustrated by an example in the next paragraph.Although in this paper we will only be concerned with the projective plane P 2 , we briefly digress to explain the fundamental difficulties with generalizing the irreducibility statement to other surfaces in the case of K3s. It is an open problem whether the Severi variety of genus h g curves in the primitive class of a general degree 2g − 2 K3 surface is irreducible. The analogue of step (3) above is known to hold by work of Chen [3,4], that is, every irreducible component of the (closure of the) Severi variety contains nodal curves of strictly smaller genus. At first glance, this makes an analogous approach by induction on h look attractive. The main problem is that the base case fails: the primitive class contains finitely many rational curves, so the genus 0 Severi variety is certainly not irreducible. If we instead try to start at h = 1, we immediately run into the problem that we lack the methods to prove this case.The bottom line is that, in order to approach the irreducibility problem in general, we need a method which avoids the reduction to the genus...

We study 'bubbling up' on nodal curves in the style of Knudsen's proof that M g,n+1 is the universal curve over Mg,n, in the presence of a (logarithmic) vector field on the nodal curve. We propose a two-step bubbling up procedure. The first step is simply Knudsen stabilization with the additional data to keep track of. The second step ensures that the vector field doesn't vanish at the inserted point. These operations work in families.As an application, we prove that the Losev-Manin compactification of the space of configurations of n points on P 1 \{0, ∞} modulo scaling degenerates isotrivially to a compactification of the space of configurations of n points on A 1 modulo translation. The latter is related to the compactification constructed by Ziltener and Mau-Woodward, but allows the marked points to coincide, making it a G n−1 a -variety, which mirrors the fact that the Losev-Manin space is toric. Its combinatorics is dual to that of the space of phylogenetic trees of Billera-Holmes-Vogtmann. The degeneration is compatible with the actions of G n−1 m and G n−1 a in the sense that these actions fit together globally.

We consider a family of quintic threefolds specializing to a certain reducible threefold. We describe the space of genus zero stable morphisms to the central fiber, as defined by J. Li. As an application of a straightforward extension, we prove the existence of rigid stable maps with smooth source of arbitrary genus and sufficiently high degree to very general quintics.