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We show that on a metric graph of genus g, a divisor of degree (n) generically has (g(n-g+1)) Weierstrass points. For a sequence of generic divisors on a metric graph whose degrees grow to infinity, we show that the associated Weierstrass points become distributed according to the Zhang canonical measure. In other words, the limiting distribution is determined by effective resistances on the metric graph. This distribution result has an analogue for complex algebraic curves, due to Neeman, and for curves over non-Archimedean fields, due to Amini.

Abstract

We prove that the moduli space of double covers ramified at two points ({mathcal {R}}_{g,2}) is uniruled for (3le gle 6) and of general type for (gge 16) . Furthermore, we consider Prym-canonical divisorial strata in the moduli space (overline{{mathcal {C}}^n{mathcal {R}}}_g) parametrizing n-pointed Prym curves, and we compute their classes in (textrm{Pic}_{mathbb {Q}}(overline{{mathcal {C}}^n{mathcal {R}}}_g)) .

This paper considers the (negative) cyclic open–closed map ({mathcal{O}mathcal{C}}^{-}), which maps the cyclic homology of the Fukaya category of a symplectic manifold to its (S^1)-equivariant quantum cohomology. We prove (under simplifying technical hypotheses) that this map respects the respective natural connections in the direction of the equivariant parameter. In the monotone setting this allows us to conclude that ({mathcal{O}mathcal{C}}^{-}) intertwines the decomposition of the Fukaya category by eigenvalues of quantum cup product with the first Chern class, with the Hukuhara–Levelt–Turrittin decomposition of the quantum cohomology. We also explain how our results relate to the Givental–Teleman classification of semisimple cohomological field theories: in particular, how the R-matrix is related to ({mathcal{O}mathcal{C}}^{-}) in the semisimple case; we also consider the non-semisimple case.

Bhargava and the first-named author of this paper introduced a functorial Galois closure operation for finite-rank ring extensions, generalizing constructions of Grothendieck and Katz–Mazur. In this paper, we generalize Galois closures and apply them to construct a new infinite family of irreducible components of Hilbert schemes of points. We show that these components are elementary, in the sense that they parametrize algebras supported at a point. Furthermore, we produce secondary families of elementary components obtained from Galois closures by modding out by suitable socle elements.

We prove the surjectivity part of Goncharov’s depth conjecture over a quadratically closed field. We also show that the depth conjecture implies that multiple polylogarithms of depth d and weight n can be expressed via a single function ({{,textrm{Li},}}_{n-d+1,1,dots ,1}(a_1,a_2,dots ,a_d)), and we prove this latter statement for (d=2).

Davydov–Yetter (DY) cohomology classifies infinitesimal deformations of the monoidal structure of tensor functors and tensor categories. In this paper we provide new tools for the computation of the DY cohomology for finite tensor categories and exact functors between them. The key point is to realize DY cohomology as relative Ext groups. In particular, we prove that the infinitesimal deformations of a tensor category ({mathcal {C}}) are classified by the 3-rd self-extension group of the tensor unit of the Drinfeld center ({mathcal {Z}}({mathcal {C}})) relative to ({mathcal {C}}). From classical results on relative homological algebra we get a long exact sequence for DY cohomology and a Yoneda product for which we provide an explicit formula. Using the long exact sequence and duality, we obtain a dimension formula for the cohomology groups based solely on relatively projective covers which reduces a problem in homological algebra to a problem in representation theory, e.g. calculating the space of invariants in a certain object of ({mathcal {Z}}({mathcal {C}})). Thanks to the Yoneda product, we also develop a method for computing DY cocycles explicitly which are needed for applications in the deformation theory. We apply these tools to the category of finite-dimensional modules over a finite-dimensional Hopf algebra. We study in detail the examples of the bosonization of exterior algebras (Lambda {mathbb {C}}^k rtimes {mathbb {C}}[{mathbb {Z}}_2]), the Taft algebras and the small quantum group of (mathfrak {sl}_2) at a root of unity.

We describe the monodromy of the equivariant quantum differential equation of the cotangent bundle of a Grassmannian in terms of the equivariant (,K,)-theory algebra of the cotangent bundle. This description is based on the hypergeometric integral representations for solutions of the equivariant quantum differential equation. We identify the space of solutions with the space of the equivariant (,K,)-theory algebra of the cotangent bundle. In particular, we show that for any element of the monodromy group, all entries of its matrix in the standard basis of the equivariant (,K,)-theory algebra of the cotangent bundle are Laurent polynomials with integer coefficients in the exponentiated equivariant parameters.

Abstract

We discuss various applications of a uniform vanishing result for the graded components of the finite length Koszul module associated to a subspace (Ksubseteq bigwedge ^2 V) , where V is a vector space. Previously Koszul modules of finite length have been used to give a proof of Green’s Conjecture on syzygies of generic canonical curves. We now give applications to effective stabilization of cohomology of thickenings of algebraic varieties, divisors on moduli spaces of curves, enumerative geometry of curves on K3 surfaces and to skew-symmetric degeneracy loci. We also show that the instability of sufficiently positive rank 2 vector bundles on curves is governed by resonance and give a splitting criterion.

Abstract

The ({textsf{SL}}(2,{{mathbb {Z}}})) -symmetry of Cherednik’s spherical double affine Hecke algebras in Macdonald theory includes a distinguished generator which acts as a discrete time evolution of Macdonald operators, which can also be interpreted as a torus Dehn twist in type A. We prove for all twisted and untwisted affine algebras of type ABCD that the time-evolved q-difference Macdonald operators, in the (trightarrow infty ) q-Whittaker limit, form a representation of the associated discrete integrable quantum Q-systems, which are obtained, in all but one case, via the canonical quantization of suitable cluster algebras. The proof relies strongly on the duality property of Macdonald and Koornwinder polynomials, which allows, in the q-Whittaker limit, for a unified description of the quantum Q-system variables and the conserved quantities as limits of the time-evolved Macdonald operators and the Pieri operators, respectively. The latter are identified with relativistic q-difference Toda Hamiltonians. A crucial ingredient in the proof is the use of the “Fourier transformed” picture, in which we compute time-translation operators and prove that they commute with the Pieri operators or Hamiltonians. We also discuss the universal solutions of Koornwinder-Macdonald eigenvalue and Pieri equations, for which we prove a duality relation, which simplifies the proofs further.

We associate to a projective n-dimensional toric variety (X_{Delta }) a pair of co-commutative (but generally non-commutative) Hopf algebras (H^{alpha }_X, H^{T}_X). These arise as Hall algebras of certain categories ({text {Coh}}^{alpha }(X), {text {Coh}}^T(X)) of coherent sheaves on (X_{Delta }) viewed as a monoid scheme—i.e. a scheme obtained by gluing together spectra of commutative monoids rather than rings. When (X_{Delta }) is smooth, the category ({text {Coh}}^T(X)) has an explicit combinatorial description as sheaves whose restriction to each (mathbb {A}^n) corresponding to a maximal cone (sigma in Delta ) is determined by an n-dimensional generalized skew shape. The (non-additive) categories ({text {Coh}}^{alpha }(X), {text {Coh}}^T(X)) are treated via the formalism of proto-exact/proto-abelian categories developed by Dyckerhoff–Kapranov. The Hall algebras (H^{alpha }_X, H^{T}_X) are graded and connected, and so enveloping algebras (H^{alpha }_X simeq U(mathfrak {n}^{alpha }_X)), (H^{T}_X simeq U(mathfrak {n}^{T}_X)), where the Lie algebras (mathfrak {n}^{alpha }_X, mathfrak {n}^{T}_X) are spanned by the indecomposable coherent sheaves in their respective categories. We explicitly work out several examples, and in some cases are able to relate (mathfrak {n}^T_X) to known Lie algebras. In particular, when (X = mathbb {P}^1), (mathfrak {n}^T_X) is isomorphic to a non-standard Borel in (mathfrak {gl}_2 [t,t^{-1}]). When X is the second infinitesimal neighborhood of the origin inside (mathbb {A}^2), (mathfrak {n}^T_X) is isomorphic to a subalgebra of (mathfrak {gl}_2[t]). We also consider the case (X=mathbb {P}^2), where we give a basis for (mathfrak {n}^T_X) by describing all indecomposable sheaves in ({text {Coh}}^T(X)).