Towards the geometry of Severi varieties

平面曲线可以说是代数几何中最经典的物体。Severi簇是参数化定次几何亏格的射影平面曲线的空间。更一般地，我们可以用其他射影曲面代替射影平面，并考虑将给定曲面上的固定同调类和几何亏格的曲线参数化的“Severi簇”。 虽然Severi簇比高维代数簇上的曲线的模空间简单得多，但人们对它的几何了解仍然很少，没有看到实质性的进展。另一方面，Severi簇的计数方面，即对射影曲面上的曲线进行计数，一直是许多研究的主题，也有很多人知道。 我的建议是深入研究一些著名的曲线计数公式背后的几何，目的是在理解Severi簇的几何性质方面取得进展。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“块”还是多个块组成的。这是我打算集中精力解决的问题。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。我希望这些技巧在未来有改进的潜力，以开始揭示Severi簇更精细的性质，并可能揭示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.