Towards the geometry of Severi varieties

走向 Severi 品种的几何形状

• 批准号: RGPIN-2020-05497
• 负责人: Zahariuc, Adrian
• 金额: $ 1.31万
• 依托单位: University of Windsor
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2022
• 资助国家: 加拿大
• 起止时间: 2022-01-01 至 2023-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=759187
• 关键词: Towards; geometry; Severi; varieties

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.

平面曲线可以说是代数几何中最经典的对象。多变种是将固定度和几何属的投影平面曲线参数化的空间。更一般地说，我们可以用其他投影曲面来代替投影平面，并考虑“Severi变种”，该变种将给定曲面上的固定同调类和几何属曲线参数化。虽然Severi变分比高维代数变分上曲线的模空间简单得多，但人们对其几何结构的理解仍然很差，看不到实质性的进展。另一方面，Severi变种的枚举方面，即对投影曲面上的曲线进行计数，一直是许多研究的主题，并且已经知道了很多。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原理，目的是在理解Severi品种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Severi变体（某些投影曲面）是否不可约，也就是说，它们是由单个“块”还是多个“块”组成。这就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。我希望这些技术有潜力在未来得到改进，开始揭示一些更精细的Severi变种的性质，以及可能在代数变种上曲线的其他模空间。