complete intersections
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2022 ◽  
pp. 1-20
Author(s):  
CRIS NEGRON ◽  
JULIA PEVTSOVA

Abstract We introduce an infinite variant of hypersurface support for finite-dimensional, noncommutative complete intersections. We show that hypersurface support defines a support theory for the big singularity category $\operatorname {Sing}(R)$ , and that the support of an object in $\operatorname {Sing}(R)$ vanishes if and only if the object itself vanishes. Our work is inspired by Avramov and Buchweitz’ support theory for (commutative) local complete intersections. In the companion piece [27], we employ hypersurface support for infinite-dimensional modules, and the results of the present paper, to classify thick ideals in stable categories for a number of families of finite-dimensional Hopf algebras.


2021 ◽  
Vol 2021 (12) ◽  
Author(s):  
Daniel Kläwer

Abstract We test the refined distance conjecture in the vector multiplet moduli space of 4D $$ \mathcal{N} $$ N = 2 compactifications of the type IIA string that admit a dual heterotic description. In the weakly coupled regime of the heterotic string, the moduli space geometry is governed by the perturbative heterotic dualities, which allows for exact computations. This is reflected in the type IIA frame through the existence of a K3 fibration. We identify the degree d = 2N of the K3 fiber as a parameter that could potentially lead to large distances, which is substantiated by studying several explicit models. The moduli space geometry degenerates into the modular curve for the congruence subgroup Γ0(N)+. In order to probe the large N regime, we initiate the study of Calabi-Yau threefolds fibered by general degree d > 8 K3 surfaces by suggesting a construction as complete intersections in Grassmann bundles.


Author(s):  
Harold Erbin ◽  
Riccardo Finotello ◽  
Robin Schneider ◽  
Mohamed Tamaazousti

Abstract We continue earlier efforts in computing the dimensions of tangent space cohomologies of Calabi-Yau manifolds using deep learning. In this paper, we consider the dataset of all Calabi-Yau four-folds constructed as complete intersections in products of projective spaces. Employing neural networks inspired by state-of-the-art computer vision architectures, we improve earlier benchmarks and demonstrate that all four non-trivial Hodge numbers can be learned at the same time using a multi-task architecture. With 30 % (80 %) training ratio, we reach an accuracy of 100 % for h(1,1) and 97 % for h(2,1) (100 % for both), 81 % (96 %) for h(3,1), and 49 % (83 %) for h(2,2). Assuming that the Euler number is known, as it is easy to compute, and taking into account the linear constraint arising from index computations, we get 100 % total accuracy.


2021 ◽  
Vol 67 (1) ◽  
pp. 1-44
Author(s):  
Brendan Hassett ◽  
Yuri Tschinkel ◽  
Jean-Louis Colliot-Thélène

Author(s):  
Nathan Ilten ◽  
Tyler L. Kelly

AbstractWe study Fano schemes $$\mathrm{F}_k(X)$$ F k ( X ) for complete intersections X in a projective toric variety $$Y\subset \mathbb {P}^n$$ Y ⊂ P n . Our strategy is to decompose $$\mathrm{F}_k(X)$$ F k ( X ) into closed subschemes based on the irreducible decomposition of $$\mathrm{F}_k(Y)$$ F k ( Y ) as studied by Ilten and Zotine. We define the “expected dimension” for these subschemes, which always gives a lower bound on the actual dimension. Under additional assumptions, we show that these subschemes are non-empty and smooth of the expected dimension. Using tools from intersection theory, we can apply these results to count the number of linear subspaces in X when the expected dimension of $$\mathrm{F}_k(X)$$ F k ( X ) is zero.


2021 ◽  
Vol 157 (9) ◽  
pp. 2046-2088
Author(s):  
Gebhard Böckle ◽  
Chandrashekhar B. Khare ◽  
Jeffrey Manning

In his work on modularity theorems, Wiles proved a numerical criterion for a map of rings $R\to T$ to be an isomorphism of complete intersections. He used this to show that certain deformation rings and Hecke algebras associated to a mod $p$ Galois representation at non-minimal level are isomorphic and complete intersections, provided the same is true at minimal level. In this paper we study Hecke algebras acting on cohomology of Shimura curves arising from maximal orders in indefinite quaternion algebras over the rationals localized at a semistable irreducible mod $p$ Galois representation $\bar {\rho }$ . If $\bar {\rho }$ is scalar at some primes dividing the discriminant of the quaternion algebra, then the Hecke algebra is still isomorphic to the deformation ring, but is not a complete intersection, or even Gorenstein, so the Wiles numerical criterion cannot apply. We consider a weight-2 newform $f$ which contributes to the cohomology of the Shimura curve and gives rise to an augmentation $\lambda _f$ of the Hecke algebra. We quantify the failure of the Wiles numerical criterion at $\lambda _f$ by computing the associated Wiles defect purely in terms of the local behavior at primes dividing the discriminant of the global Galois representation $\rho _f$ which $f$ gives rise to by the Eichler–Shimura construction. One of the main tools used in the proof is Taylor–Wiles–Kisin patching.


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