arXiv subjects

Carl Lian

Publications and source records attributed to Carl Lian.

29 records · Page 2Linked to original sources

arXiv2022

Generalized Tevelev degrees of $\mathbb{P}^1$

Let $(C,p_1,\ldots,p_n)$ be a general curve. We consider the problem of enumerating covers of the projective line by $C$ subject to incidence conditions at the marked points. These counts have been obtained by the first named author with Pandharipande and Schmitt via intersection theory on Hurwitz spaces and by the second named author with Farkas via limit linear series. In this paper, we build on these two approaches to generalize these counts to the situation where the covers are constrained to have arbitrary ramification profiles: that is, additional ramification conditions are imposed at the marked points, and some collections of marked points are constrained to have equal image.

math.AG↗

arXiv2022

Linear series on general curves with prescribed incidence conditions

Using degeneration and Schubert calculus, we consider the problem of computing the number of linear series of given degree $d$ and dimension $r$ on a general curve of genus $g$ satisfying prescribed incidence conditions at $n$ points. We determine these numbers completely for linear series of arbitrary dimension when $d$ is sufficiently large, and for all $d$ when either $r=1$ or $n=r+2$. Our formulas generalize and give new proofs of recent results of Tevelev and of Cela-Pandharipande-Schmitt.

math.AG↗

arXiv2021

Non-tautological Hurwitz cycles

We show that various loci of stable curves of sufficiently large genus admitting degree $d$ covers of positive genus curves define non-tautological algebraic cycles on $\overline{\mathcal{M}}_{g,N}$, assuming the non-vanishing of the $d$-th Fourier coefficient of a certain modular form. Our results build on those of Graber-Pandharipande and van Zelm for degree 2 covers of elliptic curves; the main new ingredient is a method to intersect the cycles in question with boundary strata, as developed recently by Schmitt-van Zelm and the author.

math.AG↗

arXiv2021

Z/rZ-equivariant covers of P^1 with moving ramification

Let X -> P^1 be a general cyclic cover. We give a simple formula for the number of equivariant meromorphic functions on X subject to ramification conditions at variable points. This generalizes and gives a new proof of a recent result of the second author and Pirola on hyperelliptic odd covers.

math.AG↗

arXiv2021

The H-tautological ring

We extend the theory of tautological classes on moduli spaces of stable curves to the more general setting of moduli spaces of admissible Galois covers of curves, introducing the so-called H-tautological ring. The main new feature is the existence of restriction-corestriction morphisms remembering intermediate quotients of Galois covers, which are a rich source of new classes. In particular, our new framework includes classes of Harris-Mumford admissible covers on moduli spaces of curves, which are known in some (and speculatively many more) examples to lie outside the usual tautological ring. We give additive generators for the H-tautological ring and show that their intersections may be algorithmically computed, building on work of Schmitt-van Zelm. As an application, we give a method for computing integrals of Harris-Mumford loci against tautological classes of complementary dimension, recovering and giving a mild generalization of a recent quasi-modularity result of the author for covers of elliptic curves.

math.AG↗

arXiv2020

An asymptotic Alexander-Hirschowitz theorem for surfaces

Let X be a smooth projective surface over C and let L be an ample line bundle on X. In this note, we show that, for all sufficiently large d, any number of general double points on X imposes the expected number of conditions on the linear system |L^d|. Equivalently, the space of d-plane sections of X singular at any number of general points has the expected dimension. We conjecture that the same holds for X of arbitrary dimension.

math.AG↗

arXiv2020

Enumerating pencils with moving ramification on curves

We consider the general problem of enumerating branched covers of the projective line from a fixed general curve subject to ramification conditions at possibly moving points. Our main computations are in genus 1; the theory of limit linear series allows one to reduce to this case. We first obtain a simple formula for a weighted count of pencils on a fixed elliptic curve E, where base-points are allowed. We then deduce, using an inclusion-exclusion procedure, formulas for the numbers of maps E->P^1 with moving ramification conditions. A striking consequence is the invariance of these counts under a certain involution. Our results generalize work of Harris, Logan, Osserman, and Farkas-Moschetti-Naranjo-Pirola.

math.AG↗

arXiv2020

The class of the d-elliptic locus in genus 2

We compute the rational Chow class of the locus of genus 2 curves admitting a d-to-1 map to a genus 1 curve, recovering a result of Faber-Pagani when d=2. The answer exhibits quasi-modularity properties similar to those in the Gromov-Witten theory of a fixed genus 1 curve. Along the way, we give a classification of Harris-Mumford admissible covers of a genus 1 curve by a genus 2 curve, which may be of independent interest. As an application of the main calculation, we compute the number of d-elliptic curves in a very general family of genus 2 curves obtained by fixing five branch points of the hyperelliptic map and varying the sixth.

math.AG↗

arXiv2014

Representations of Cherednik Algebras Associated to Symmetric and Dihedral Groups in Positive Characteristic

We consider irreducible lowest-weight representations of Cherednik algebras associated to certain classes of complex reflection groups in characteristic p. In particular, we study maximal graded submodules of Verma modules associated to these algebras. Various results and conjectures are presented concerning generators of these maximal submodules, which are found by computing singular polynomials of Dunkl operators.

math.RT↗

arXiv2013

Circular Planar Electrical Networks II: Positivity Phenomena

Curtis-Ingerman-Morrow characterize response matrices for circular planar electrical networks as symmetric square matrices with row sums zero and non-negative circular minors. In this paper, we study this positivity phenomenon more closely, from both algebraic and combinatorial perspectives. Extending work of Postnikov, we introduce electrical positroids, which are the sets of circular minors which can simultaneously be positive in a response matrix. We give a self-contained axiomatic description of these electrical positroids. In the second part of the paper, we discuss a naturally arising example of a Laurent phenomenon algebra, as studied by Lam-Pylyavskyy. We investigate the clusters in this algebra, building off of initial work by Kenyon-Wilson, using an analogue of weak separation, as was originally introduced by Leclerc-Zelevinsky.

math.CO↗

arXiv2013

Circular Planar Electrical Networks I: The Electrical Poset EP_{n}

Following de Verdière-Gitler-Vertigan and Curtis-Ingerman-Morrow, we prove a host of new results on circular planar electrical networks. We introduce a poset EP_{n} of electrical networks with n boundary vertices, giving two equivalent characterizations, one combinatorial and the other topological. We then investigate various properties of the EP_{n}, proving that it is graded by number of edges of critical representatives. Finally, we answer various enumerative questions related to EP_{n}, adapting methods of Callan and Stein-Everett.

math.CO↗