Classical Algebraic Geometry
Classical Algebraic Geometry
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DOI：
10.4171/owr/2010/27
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发表时间：
2010
期刊：
影响因子：
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通讯作者：
C. Voisin;D. Erman
中科院分区：
文献类型：
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作者：
C. Voisin;D. Erman
代数几何一方面研究特定代数变体的性质，另一方面研究所有固定拓扑类型变体的模空间。其中特别重要的是曲线的模空间，其性质一直是研究的主题。这些和相关空间的合理性与一般类型的问题是经典的，也是非常现代的兴趣，最近在会议上提出了进展。曲线和映射的模空间的某些不同的二元模型可以解释为奇异曲线和映射的模空间。对于特定的品种，广泛的问题被解决，包括外在的问题（合子，k-割线引理）和内在的问题（线束的正性概念的推广，理想和束的闭包操作）。数学学科分类（2000）：14xx。由David Eisenbud（伯克利）、Frank-Olaf Schreyer（萨尔布尔<s:1>肯大学）、Ravi Vakil（斯坦福大学）和Claire Voisin（巴黎）主办的经典代数几何研讨会于2010年6月20日至26日在Oberwolfach数学研究所举行。来自美国、加拿大、英国、意大利、法国、波兰和德国的50多名与会者出席了会议。总共有18个一小时的讲座，每天最多4个讲座，还有一个简短的演讲环节，让年轻的参与者用10分钟的时间概述他们目前的工作。这个时间表为许多非正式讨论和小组工作留下了足够的空间。扩展摘要详细描述了会议的各种主题，其中许多是用现代方法讨论的代数几何中的经典问题。虽然我们可以在这里提及所有的演讲，但我们应该关注几个亮点：•会议以加夫里尔·法卡斯的演讲开始，他的主要新工作是格林关于正则曲线的合性猜想。之前的重大突破是Voisin在本世纪初的证明，证明Green猜想适用于秩为1的Picard群的K3曲面上的光滑曲线，以及Aprodu的结果（建立在Voisin的基础上），证明Green猜想适用于所有满足线性增长条件的曲线，从而将Green猜想变成了Brill-Noether理论中的问题。法卡斯解释了他最近的证明，证明格林的猜想实际上适用于任意K3曲面上的每一条光滑曲线。•丹尼尔·埃尔曼（Daniel Erman），在来之前不久获得了博士学位，他报告了与梅勒妮·伍德（Melanie Wood）（另一位最近获得博士学位的人，AIM的五年制研究员）的联合研究。他们的工作涉及到n次的空间覆盖的变化（或方案），他们用点的模的空间（堆栈）优雅地解释。如果n≤5，包含矩阵表示（或其几何变化）的漂亮描述会导致明确的有用结果。（一个突出的例子是曼朱尔·巴尔加瓦（Manjul Bhargava）在数论方面的著名著作。）他们开始讨论案例n = 6，弄清楚是什么让这个案例变得困难。一种特殊的情况是由Gulliksen-Negard复合体产生的构型，它们对由这种构造产生的性盖类给出了重要的限制。•热带几何的最新发展对代数几何产生了重大影响，它将许多经典问题转化为可用于计算或证明定理的分段线性问题。Sam Payne报告了最近与Brian Osserman在热带几何中发展交集理论的工作，以这样一种方式，结果将转化为经典代数几何。这需要系统地发展维数理论和非经典对象上的交集理论：秩1的非noether估值环上的有限型方案。他们的一般理论承诺包含和扩展早期的特别方法，并将实质性地改变主题。•乔·哈里斯（Joe Harris）的结束语将曲线参数空间的双空间几何的一些最新进展联系在一起。曲线空间的不同紧化（自身紧化和射影空间紧化）在许多情况下被证明是有用的。最近（首先由Hassett和Keel）观察到，这些不同的紧化通常以几何上有意义的方式联系在一起。例如，D. Chen， Coskun和Crissman已经证明了在射影空间中许多有理曲线的空间都是以这种方式产生的。此外，Viscardi还推广了Smyth等人最近关于1属曲线的工作，以发现1属曲线在射光空间中的小紧化，经典代数几何1575可能希望这与Marian， Oprea和Pandharipande最近的“稳定对”构造有关。来自Michele Bolognesi、Dawei Chen、Christian Christensen、Thomas Dedieu、Florian Geiss、Andreas Höring、Grzegorz Kapustka、Paul Larsen和Margherita Lelli-Chiesa的年轻与会者的演讲也涵盖了从曲线模量、矢量束、K3曲面到阿贝列变换的广泛主题。
Algebraic geometry studies properties of specific algebraic varieties, on the one hand, and moduli spaces of all varieties of fixed topological type on the other hand. Of special importance is the moduli space of curves, whose properties are subject of ongoing research. The rationality versus general type question of these and related spaces is of classical and also very modern interest with recent progress presented in the conference. Certain different birational models of the moduli space of curves and maps have an interpretation as moduli spaces of singular curves and maps. For specific varieties a wide range of questions was addressed, including extrinsic questions (syzygies, the k-secant lemma) and intrinsic ones (generalization of notions of positivity of line bundles, closure operations on ideals and sheaves). Mathematics Subject Classification (2000): 14xx. Introduction by the Organisers The workshop Classical Algebraic Geometry held from June 20th to June 26th 2010 at the ”Mathematisches Forschungsinstitut Oberwolfach” was organized by David Eisenbud (Berkeley), Frank-Olaf Schreyer (Saarbrücken), Ravi Vakil (Stanford) and Claire Voisin (Paris). It was very well attended with over 50 participants from the United States, Canada, United Kingdom, Italy, France, Poland and Germany. In total there were 18 one hour talks with a maximum of four talks a day and a session with short presentations allowing young participants to give 10 minute outlines of their current work. This schedule left plenty of room for many informal discussions and work in smaller groups. 1574 Oberwolfach Report 27/2010 The extended abstracts give a detailed account of the broad variety of topics of the meeting, many of them classical questions in algebraic geometry discussed with modern methods. Although it would be nice if we could mention all talks here, we should focus on a couple of highlights: • The conference opened with Gavril Farkas’ lecture on his major new work on Green’s conjecture on syzygies of canonical curves. The previous major breakthrough was Voisin’s proof earlier this decade that Green’s conjecture holds for smooth curves lying on a K3 surface with Picard group of rank 1, and Aprodu’s result (building on Voisin) that Green’s conjecture holds for all curves satisfying the linear growth condition, thus turning Green’s conjecture into a question in Brill-Noether theory. Farkas explained his recent proof that Green’s conjecture in fact holds for every smooth curve lying on an arbitrary K3 surface. • Daniel Erman, who received his Ph.D. shortly before arriving, reported on joint work with Melanie Wood (another recent Ph.D., and an AIM five-year-fellow). Their work deals with the space of degree n covers of varieties (or schemes), which they elegantly interpret in terms of the space (stack) of moduli of points. If n ≤ 5, beautiful descriptions involving matrix presentations (or geometric variations thereof) lead to explicit useful results. (One prominent example is the celebrated work in number theory of Manjul Bhargava.) They begin to address the case n = 6, making clear what makes this case difficult. A special case are configurations arising from the Gulliksen-Negard complex, and they give nontrivial restrictions on the class of sextic covers which can arise from such a construction. • The recent development of tropical geometry has had a significant impact on algebraic geometry, by turning many classical problems into piecewise linear problems that one can work with, either to calculate, or to prove theorems. Sam Payne reported on recent work with Brian Osserman developing intersection theory in tropical geometry, in such a way that the results will translate into classical algebraic geometry. This requires the systematic development of dimension theory and intersection theory on decidedly non-classical objects: schemes of finite type over non-Noetherian valuation rings of rank 1. Their general theory promises to subsume and extend earlier ad hoc methods, and will substantially change the subject. • The concluding lecture of Joe Harris tied together a number of recent advances on the birational geometry of parameter spaces of curves. Different compactifications of the space of curves (both by themselves, and in projective space) have proved useful in a number of contexts. It has been recently observed (first by Hassett and Keel) that these different compactifications are often related in geometrically meaningful ways. For example, D. Chen, Coskun, and Crissman have shown that many of the spaces of rational curves in projective space arise in this way. Also, Viscardi has generalized recent work of Smyth and others on genus 1 curves to find a small compactification of genus 1 curves in projective space, which one Classical Algebraic Geometry 1575 might hope is related to the recent “stable pairs” construction of Marian, Oprea, and Pandharipande. The young participants’ presentations by Michele Bolognesi, Dawei Chen, Christian Christensen, Thomas Dedieu, Florian Geiss, Andreas Höring, Grzegorz Kapustka, Paul Larsen and Margherita Lelli-Chiesa also covered a widespread variety of topics from moduli of curves, vector bundles, K3 surfaces to abelian varieties.