Towards the geometry of Severi varieties

平面曲线可以说是代数几何中最经典的对象。Severi簇是一个空间，它参数化了固定次数和几何亏格的射影平面曲线。更一般地说，我们可以用其他的投影曲面代替投影平面，并考虑“塞维里簇”，它参数化给定曲面上的固定同调类和几何亏格的曲线。 虽然塞维里簇比高维代数簇上的曲线的模空间简单得多，但它们的几何仍然很难理解，没有实质性的进展。另一方面，枚举方面的塞维品种，即计数曲线的投影曲面，一直是主题的大量调查和很多是众所周知的。我的建议是深入调查的几何基础的一些著名的曲线计数公式，取得进展的目标是了解几何性质的塞维品种。在深入研究更微妙的性质之前，需要澄清的第一个问题是（某些射影曲面的）Severi簇是否是不可约的，也就是说，它们是由单个“片”还是多个片组成。这是我打算集中研究的问题，我打算使用的方法属于代数几何中经常使用的一类技术，称为退化方法。我希望这些技术在未来有可能得到改进，开始阐明Severi簇的更精细的性质，可能还有代数簇上曲线的其他模空间。

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.