ABELIAN VARIETIES, MODULI OF VECTOR BUNDLES AND MODULI OF COURVES

在与G. Pareschi的合作中，Popa引入了用傅里叶-Mukai变换定义的阿贝尔变体上相干束的Mukai正则性概念，并发展了它的理论。Popa和Pareschi现在热衷于将Mukai正则性的概念应用于不规则变体的研究，特别是在伴随线性序列和多正则映射上给出了有效的结果。此外，Popa和Pareschi打算使用从这个理论中自然产生的不变量来区分雅可比矩阵和其他阿贝尔变量，从而以一种新的方式处理肖特基问题。利用他们在任意傅里叶变换的更一般背景下的一些结果，Popa和Pareschi打算获得派生类别等价存在的约束，这是这条镜像对称方法的主要关注点。在其他工作中，Popa在曲线上向量束的模空间上建立了线性级数的有效结果。Popa基于雅可比矩阵上Verlinde束的Fourier-Mukai变换并将其与表示理论技术相结合，通过改进他的方法来研究奇异对偶性。利用退化到稳定曲线的方法，探讨了一般曲线在这些模空间上的有效基点自由度的一些最优猜想。在与G. Farkas的合作中，Popa利用可约曲线上的广义极限线性级数和对的稳定性，得到了一般曲线上2阶向量束的Brill-Noether型不存在性结果。Popa和Farkas打算进一步完善这些技术，以证明bertram - feinberg - mukai在Brill-Noether理论上对2阶正则行列式向量束的一个猜想，并找到更高阶的不存在结果。这将允许他们在稳定曲线的模空间中为适当的属定义新的除数。该项目的最终目标是计算这些除数的类别，并确定它们是否如预期的那样为哈里斯-莫里森斜率猜想提供反例。代数曲线和阿贝尔变数是数学中普遍存在的对象。除了在代数几何中实现外，它们还出现在复杂分析（如黎曼曲面或复环面）或代数（如场扩展或群方案）中，并在数学物理的最新进展中发挥了重要作用。代数几何中最重要的问题之一是代数变量的分类。对于一维变量（即代数曲线），这个问题是通过理解空间M(g)的几何形状来参数化给定属的所有曲线来解决的。在阿贝尔变体的情况下，解决这个问题的一种方法是理解与它们相关的特定代数对象（称为相干束）的总体。研究者在与本提案相关的研究中正在追求这些方向。

In work with G. Pareschi, Popa has introduced a notion of Mukai regularity for coherent sheaves on abelian varieties, defined using the Fourier-Mukai transform, and developed its theory. Popa and Pareschi are now interested in using the concept of Mukai regularity in the study of irregular varieties, especially to give effective results on adjoint linear series and pluiricanonical maps. Also, Popa and Pareschi intend to use invariants arising naturally from this theory in order to differentiate between Jacobians and other abelian varieties, and thus approach the Schottky problem in a new way. Using some of their results that carry through in the more general context of an arbitrary Fourier transform, Popa and Pareschi intend to obtain constraints on the existence of equivalences of derived categories, which is a main concern in this line of approach to mirror symmetry. In other work, Popa has established effective results for linear series on moduli spaces of vector bundles on curves. Popa is interested in approaching the Strange Duality by refining his methods based on the Fourier-Mukai transform of Verlinde bundles on Jacobians and combining it with representation theory techniques. Popa also intends to approach some optimal conjectures on effective base point freeness on these moduli spaces for general curves, using degeneration to stable curves. In work with G. Farkas, Popa has obtained Brill-Noether type non-existence results for rank 2 vector bundles on general curves, via generalized limit linear series and stability of pairs on reducible curves. Popa and Farkas intend to further refine these techniques in order to prove a conjecture of Bertram-Feinberger-Mukai on Brill-Noether theory for rank 2 vector bundles with canonical determinant, and to find non-existence results in higher ranks. This will allow them to define new divisors in moduli spaces of stable curves for appropriate genera. The ultimate goal of this project is to compute the class of these divisors and determine whether they provide counterexamples to the Harris-Morrison Slope Conjecture as expected. Algebraic curves and abelian varieties are ubiquitous objects in mathematics. Apart from their realizations in algebraic geometry, they appear in complex analysis (as Riemann surfaces or complex tori) or algebra (as field extensions or group schemes), and play a fundamental role in recent progress in mathematical physics. One of the most important problems in algebraic geometry is to classify algebraic varieties. For one-dimensional varieties (that is, for algebraic curves) this problem is approached by understanding the geometry of a space M(g) parametrizing all curves of given genus. In the case of abelian varieties, one way to approach this is to understand the totality of a specific kind of algebraic objects (called coherent sheaves) which can be associated to them. The investigator is pursuing these directions in research related to this proposal.