Let
We complete the computation of all
- NSF-PAR ID:
- 10373621
- Publisher / Repository:
- Springer Science + Business Media
- Date Published:
- Journal Name:
- Research in Number Theory
- Volume:
- 8
- Issue:
- 4
- ISSN:
- 2522-0160
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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Abstract be an elliptically fibered$$X\rightarrow {{\mathbb {P}}}^1$$ K 3 surface, admitting a sequence of Ricci-flat metrics collapsing the fibers. Let$$\omega _{i}$$ V be a holomorphicSU (n ) bundle overX , stable with respect to . Given the corresponding sequence$$\omega _i$$ of Hermitian–Yang–Mills connections on$$\Xi _i$$ V , we prove that, ifE is a generic fiber, the restricted sequence converges to a flat connection$$\Xi _i|_{E}$$ . Furthermore, if the restriction$$A_0$$ is of the form$$V|_E$$ for$$\oplus _{j=1}^n{\mathcal {O}}_E(q_j-0)$$ n distinct points , then these points uniquely determine$$q_j\in E$$ .$$A_0$$ -
Abstract We prove that the Hilbert scheme of
k points on ($${\mathbb {C}}^2$$ ) is self-dual under three-dimensional mirror symmetry using methods of geometry and integrability. Namely, we demonstrate that the corresponding quantum equivariant K-theory is invariant upon interchanging its Kähler and equivariant parameters as well as inverting the weight of the$$\hbox {Hilb}^k[{\mathbb {C}}^2]$$ -action. First, we find a two-parameter family$${\mathbb {C}}^\times _\hbar $$ of self-mirror quiver varieties of type A and study their quantum K-theory algebras. The desired quantum K-theory of$$X_{k,l}$$ is obtained via direct limit$$\hbox {Hilb}^k[{\mathbb {C}}^2]$$ and by imposing certain periodic boundary conditions on the quiver data. Throughout the proof, we employ the quantum/classical (q-Langlands) correspondence between XXZ Bethe Ansatz equations and spaces of twisted$$l\longrightarrow \infty $$ -opers. In the end, we propose the 3d mirror dual for the moduli spaces of torsion-free rank-$$\hbar $$ N sheaves on with the help of a different (three-parametric) family of type A quiver varieties with known mirror dual.$${\mathbb {P}}^2$$ -
Abstract In this article, we study the moduli of irregular surfaces of general type with at worst canonical singularities satisfying
, for any even integer$$K^2 = 4p_g-8$$ . These surfaces also have unbounded irregularity$$p_g\ge 4$$ q . We carry out our study by investigating the deformations of the canonical morphism , where$$\varphi :X\rightarrow {\mathbb {P}}^N$$ is a quadruple Galois cover of a smooth surface of minimal degree. These canonical covers are classified in Gallego and Purnaprajna (Trans Am Math Soc 360(10):5489-5507, 2008) into four distinct families, one of which is the easy case of a product of curves. The main objective of this article is to study the deformations of the other three, non trivial, unbounded families. We show that any deformation of$$\varphi $$ factors through a double cover of a ruled surface and, hence, is never birational. More interestingly, we prove that, with two exceptions, a general deformation of$$\varphi $$ is two-to-one onto its image, whose normalization is a ruled surface of appropriate irregularity. We also show that, with the exception of one family, the deformations of$$\varphi $$ X are unobstructed even though does not vanish. Consequently,$$H^2(T_X)$$ X belongs to a unique irreducible component of the Gieseker moduli space. These irreducible components are uniruled. As a result of all this, we show the existence of infinitely many moduli spaces, satisfying the strict Beauville inequality , with an irreducible component that has a proper quadruple sublocus where the degree of the canonical morphism jumps up. These components are above the Castelnuovo line, but nonetheless parametrize surfaces with non birational canonical morphisms. The existence of jumping subloci is a contrast with the moduli of surfaces with$$p_g > 2q-4$$ , studied by Horikawa. Irreducible moduli components with a jumping sublocus also present a similarity and a difference to the moduli of curves of genus$$K^2 = 2p_g- 4$$ , for, like in the case of curves, the degree of the canonical morphism goes down outside a closed sublocus but, unlike in the case of curves, it is never birational. Finally, our study shows that there are infinitely many moduli spaces with an irreducible component whose general elements have non birational canonical morphism and another irreducible component whose general elements have birational canonical map.$$g\ge 3$$ -
Abstract Let us fix a prime
p and a homogeneous system ofm linear equations for$$a_{j,1}x_1+\dots +a_{j,k}x_k=0$$ with coefficients$$j=1,\dots ,m$$ . Suppose that$$a_{j,i}\in \mathbb {F}_p$$ , that$$k\ge 3m$$ for$$a_{j,1}+\dots +a_{j,k}=0$$ and that every$$j=1,\dots ,m$$ minor of the$$m\times m$$ matrix$$m\times k$$ is non-singular. Then we prove that for any (large)$$(a_{j,i})_{j,i}$$ n , any subset of size$$A\subseteq \mathbb {F}_p^n$$ contains a solution$$|A|> C\cdot \Gamma ^n$$ to the given system of equations such that the vectors$$(x_1,\dots ,x_k)\in A^k$$ are all distinct. Here,$$x_1,\dots ,x_k\in A$$ C and are constants only depending on$$\Gamma $$ p ,m andk such that . The crucial point here is the condition for the vectors$$\Gamma in the solution$$x_1,\dots ,x_k$$ to be distinct. If we relax this condition and only demand that$$(x_1,\dots ,x_k)\in A^k$$ are not all equal, then the statement would follow easily from Tao’s slice rank polynomial method. However, handling the distinctness condition is much harder, and requires a new approach. While all previous combinatorial applications of the slice rank polynomial method have relied on the slice rank of diagonal tensors, we use a slice rank argument for a non-diagonal tensor in combination with combinatorial and probabilistic arguments.$$x_1,\dots ,x_k$$ -
Abstract We study the family of irreducible modules for quantum affine
whose Drinfeld polynomials are supported on just one node of the Dynkin diagram. We identify all the prime modules in this family and prove a unique factorization theorem. The Drinfeld polynomials of the prime modules encode information coming from the points of reducibility of tensor products of the fundamental modules associated to{\mathfrak{sl}_{n+1}} with{A_{m}} . These prime modules are a special class of the snake modules studied by Mukhin and Young. We relate our modules to the work of Hernandez and Leclerc and define generalizations of the category{m\leq n} . This leads naturally to the notion of an inflation of the corresponding Grothendieck ring. In the last section we show that the tensor product of a (higher order) Kirillov–Reshetikhin module with its dual always contains an imaginary module in its Jordan–Hölder series and give an explicit formula for its Drinfeld polynomial. Together with the results of [D. Hernandez and B. Leclerc,A cluster algebra approach to{\mathscr{C}^{-}} q -characters of Kirillov–Reshetikhin modules,J. Eur. Math. Soc. (JEMS) 18 2016, 5, 1113–1159] this gives examples of a product of cluster variables which are not in the span of cluster monomials. We also discuss the connection of our work with the examples arising from the work of [E. Lapid and A. Mínguez,Geometric conditions for -irreducibility of certain representations of the general linear group over a non-archimedean local field,Adv. Math. 339 2018, 113–190]. Finally, we use our methods to give a family of imaginary modules in type\square which do not arise from an embedding of{D_{4}} with{A_{r}} in{r\leq 3} .{D_{4}}