Towards the geometry of Severi varieties

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Plane curves are arguably the most classical objects in algebraic geometry. Severi varieties are spaces which parametrize projective plane curves of fixed degree and geometric genus. More generally, we may replace the projective plane with other projective surfaces and consider the "Severi variety" which parametrizes curves of a fixed homology class and geometric genus on the given surface. Although Severi varieties are much simpler than the moduli spaces of curves on higher dimensional algebraic varieties, their geometry is still very poorly understood, with no substantial progress in sight. On the other hand, the enumerative side of Severi varieties, that is, counting curves on projective surfaces, has been the subject of much investigation and a lot is known. My proposal is to investigate in depth the geometry underlying some well-known curve counting formulas, with the goal of making progress towards understanding the geometric properties of Severi varieties. The first question that needs to be clarified before diving into the more subtle properties is whether or not the Severi varieties (of certain projective surfaces) are irreducible, that is, whether they consist of a single "piece" or multiple pieces. This is the problem I intend to concentrate on. The methods I intend to use belong to a class of techniques frequently used in algebraic geometry, called degeneration methods. I hope these techniques have the potential to be improved in the future, to start shedding some light on the more refined properties of Severi varieties, and possibly of other moduli spaces of curves on algebraic varieties.