An official website of the United States government
The local sum conjecture is a variant of some of Igusa’s questions on exponential sums put forward by Denef and Sperber. In a remarkable paper by Cluckers, Mustata and Nguyen, this conjecture has been established in all dimensions, using sophisticated, powerful techniques from a research area blending algebraic geometry with ideas from logic. The purpose of this paper is to give an elementary proof of this conjecture in two dimensions which follows Varčenko’s treatment of Euclidean oscillatory integrals based on Newton polyhedra for good coordinate choices. Another elementary proof is given by Veys from an algebraic geometric perspective.
more »
« less
Wall-Crossing for Newton–Okounkov Bodies and the Tropical Grassmannian
Abstract Tropical geometry and the theory of Newton–Okounkov bodies are two methods that produce toric degenerations of an irreducible complex projective variety. Kaveh and Manon showed that the two are related. We give geometric maps between the Newton–Okounkov bodies corresponding to two adjacent maximal-dimensional prime cones in the tropicalization of $$X$$. Under a technical condition, we produce a natural “algebraic wall-crossing” map on the underlying value semigroups (of the corresponding valuations). In the case of the tropical Grassmannian $Gr(2,m)$, we prove that the algebraic wall-crossing map is the restriction of a geometric map. In an appendix by Nathan Ilten, he explains how the geometric wall-crossing phenomenon can also be derived from the perspective of complexity-one $$T$$-varieties; Ilten also explains the connection to the “combinatorial mutations” studied by Akhtar–Coates–Galkin–Kasprzyk.
more »
« less
- Award ID(s):
- 1855598
- PAR ID:
- 10253243
- Date Published:
- Journal Name:
- International Mathematics Research Notices
- ISSN:
- 1073-7928
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
Bauer, Tomer; Liu, Gaku (Ed.)There is an abundance of deep literature on the use of free resolutions to study modules and vector bundle resolutions to study coherent sheaves. When studying a module over the Cox ring of a smooth projective toric variety X, each approach comes with its own challenges. There is geometric information that free resolutions fail to encode, while vector bundle resolutions resist study using algebraic and combinatorial techniques. Recently, Berkesch, Erman, and Smith introduced virtual resolutions, which are amenable to algebraic and combinatorial study and also capture desirable geometric information. In this extended abstract, we continue this program in the combinatorially-rich Stanley–Reisner setting. In particular, when X is a product of projective spaces, we produce a large new class of virtually Cohen–Macaulay Stanley–Reisner rings. After augmenting the simplicial complexes associated to these Stanley–Reisner rings with a coloring that reflects the product structure on X, our primary tool is Reisner’s criterion, whose conclusion we interpret in the virtual setting. We also provide two constructions of short virtual resolutions for use beyond the Stanley–Reisner case.more » « less
-
null (Ed.)The Ceresa cycle is an algebraic cycle attached to a smooth algebraic curve with a marked point, which is trivial when the curve is hyperelliptic with a marked Weierstrass point. The image of the Ceresa cycle under a certain cycle class map provides a class in étale cohomology called the Ceresa class. Describing the Ceresa class explicitly for non-hyperelliptic curves is in general not easy. We present a "combinatorialization" of this problem, explaining how to define a Ceresa class for a tropical algebraic curve, and also for a topological surface endowed with a multiset of commuting Dehn twists (where it is related to the Morita cocycle on the mapping class group). We explain how these are related to the Ceresa class of a smooth algebraic curve over ℂ((t)), and show that the Ceresa class in each of these settings is torsion.more » « less
-
null (Ed.)CBMS notes from a series of 10 Lectures by David Cox, with supplemental lectures by C. D'Andrea, A. Dickenstein, J. Hauenstein, H. Schenck, J. Sidman. Systems of polynomial equations can be used to model an astonishing variety of phenomena. This book explores the geometry and algebra of such systems and includes numerous applications. The book begins with elimination theory from Newton to the twenty-first century and then discusses the interaction between algebraic geometry and numerical computations, a subject now called numerical algebraic geometry. The final three chapters discuss applications to geometric modeling, rigidity theory, and chemical reaction networks in detail. Each chapter ends with a section written by a leading expert.more » « less
-
null (Ed.)3D point cloud completion has been a long-standing challenge at scale, and corresponding per-point supervised training strategies suffered from cumbersome annotations. 2D supervision has recently emerged as a promising alternative for 3D tasks, but specific approaches for 3D point cloud completion still remain to be explored. To overcome these limitations, we propose an end-to-end method that directly lifts a single depth map to a completed point cloud. With one depth map as input, a multi-way novel depth view synthesis network (NDVNet) is designed to infer coarsely completed depth maps under various viewpoints. Meanwhile, a geometric depth perspective rendering module is introduced to utilize the raw input depth map to generate a reprojected depth map for each view. Therefore, the two parallelly generated depth maps for each view are further concatenated and refined by a depth completion network (DCNet). The final completed point cloud is fused from all refined depth views. Experimental results demonstrate the effectiveness of our proposed approach composed of aforementioned components, to produce high-quality, state-of-the-art results on the popular SUNCG benchmark.more » « less
- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1093/imrn/rnaa230
