arXiv subjects

Karl Christ

Publications and source records attributed to Karl Christ.

12 recordsLinked to original sources

arXiv2026

Scrollar invariants of singular curves on toric surfaces

Given a curve on a toric surface, a monomial projection induces a map from the normalization of the curve to the projective line. We determine the associated scrollar invariants for general integral curves of fixed geometric genus for a large class of toric surfaces. This generalizes a combinatorial formula to calculate such scrollar invariants for smooth curves in characteristic zero due to Castryck and Cools. We describe an expected behaviour for any toric surface, but provide examples where this fails at least on some irreducible component of the corresponding Severi variety.

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arXiv2025

The irreducibility of Hurwitz spaces and Severi varieties on toric surfaces

In 1969, Fulton introduced classical Hurwitz spaces parametrizing simple d-sheeted coverings of the projective line in the algebro-geometric setting. He established the irreducibility of these spaces under the assumption that the characteristic of the ground field is greater than d, but the irreducibility problem in smaller characteristics remained open. We resolve this problem in the current paper and prove that the classical Hurwitz spaces are irreducible over any algebraically closed field. On the way, we establish the irreducibility of Severi varieties in arbitrary characteristic for a rich class of toric surfaces, including all classical toric surfaces. Our approach to the irreducibility problems comes from tropical geometry, and the paper contains two more results of independent interest - a lifting result for parametrized tropical curves and a strong connectedness property of the moduli spaces of parametrized tropical curves.

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arXiv2024

Clifford representatives via the uniform algebraic rank

In this paper, we introduce the uniform algebraic rank of a divisor class on a finite graph. We show that it lies between Caporaso's algebraic rank and the combinatorial rank of Baker and Norine. We prove the Riemann-Roch theorem for the uniform algebraic rank, and show that both the algebraic and the uniform algebraic rank are realized on effective divisors. As an application, we use the uniform algebraic rank to show that Clifford representatives always exist. We conclude with an explicit description of such Clifford representatives for a large class of graphs.

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arXiv2024

Balancing properties of tropical moduli maps

Given a family of parameterized algebraic curves over a strictly semistable pair, we show that the simultaneous tropicalization of the curves in the family forms a family of parameterized tropical curves over the skeleton of the strictly semistable pair. We show that the induced tropical moduli map satisfies a certain balancing condition, which allows us to describe properties of its image and deduce a new liftability criterion.

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arXiv2023

Special divisors in special divisor classes on graphs

As in algebraic geometry, an effective divisor class on a vertex-weighted graph is called special if also its residual class is effective. We study the question, when this is true already on the level of divisors; that is, when there exists an effective divisor in the class whose residual is effective as well, so called uniform divisors. We show that uniform divisors exist in any special class on graphs with all vertex weights non-zero; we show that in general a representative can be chosen, whose value is off by at most one on vertices of weight zero. To prove this, we generalize the notion of divisors that are reduced with respect to a vertex to divisors that are reduced with respect to a set of vertices. We compare this notion to a previous generalization due to Luo. As an application, we obtain a new algorithm to determine whether a divisor class is effective.

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arXiv2023

Bounding the number of graph refinements for Brill-Noether existence

Let $G$ be a finite graph of genus $g$. Let $d$ and $r$ be non-negative integers such that the Brill-Noether number is non-negative. It is known that for some $k$ sufficiently large, the $k$-th homothetic refinement $G^{(k)}$ of $G$ admits a divisor of degree $d$ and rank at least $r$. We use results from algebraic geometry to give an upper bound for $k$ in terms of $g,d,$ and $r$.

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arXiv2022

A Clifford inequality for semistable curves

Let $X$ be a semistable curve and $L$ a line bundle whose multidegree is uniform, i.e., in the range between those of the structure sheaf and the dualizing sheaf of $X$. We establish an upper bound for $h^0(X,L)$, which generalizes the classic Clifford inequality for smooth curves. The bound depends on the total degree of $L$ and connectivity properties of the dual graph of $X$. It is sharp, in the sense that on any semistable curve there exist line bundles with uniform multidegree that achieve the bound.

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arXiv2020

Degeneration of curves on some polarized toric surfaces

We address the following question: Given a polarized toric surface (S,L), and a general integral curve C of geometric genus g in the linear system |L|, do there exist degenerations of C in |L| to general integral curves of smaller geometric genera? We give an affirmative answer to this question for surfaces associated to h-transverse polygons, provided that the characteristic of the ground field is large enough. We give examples of surfaces in small characteristic, for which the answer to the question is negative. In case the answer is affirmative, we deduce that a general curve C as above is nodal. In characteristic 0, we use the result to show irreducibility of Severi varieties of a large class of polarized toric surfaces with h-transverse polygon.

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arXiv2020

On the rank of general linear series on stable curves

We study the dimension of loci of special line bundles on stable curves and for a fixed semistable multidegree. In case of total degree $d = g - 1$, we characterize when the effective locus gives a Theta divisor. In case of degree $g - 2$ and $g$, we show that the locus is either empty or has the expected dimension. This leads to a new characterization of semistability in these degrees. In the remaining cases, we show that the special locus has codimension at least $2$. If the multidegree in addition is non-negative on each irreducible component of the curve, we show that the special locus contains an irrreducible component of expected dimension.

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arXiv2020

On the Severi problem in arbitrary characteristic

In this paper, we show that Severi varieties parameterizing irreducible reduced planar curves of a given degree and geometric genus are either empty or irreducible in any characteristic. Following Severi's original idea, this gives a new proof of the irreducibility of the moduli space of smooth projective curves of a given genus in positive characteristic. It is the first proof that involves no reduction to the characteristic zero case. As a further consequence, we generalize Zariski's theorem to positive characteristic and show that a general reduced planar curve of a given geometric genus is nodal.

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arXiv2019

Compactified Jacobians as Mumford models

We show that relative compactified Jacobians of one-parameter smoothings of a nodal curve of genus g are Mumford models of the generic fiber. Each such model is given by an admissible polytopal decomposition of the skeleton of the Jacobian. We describe the decompositions corresponding to compactified Jacobians explicitly in terms of the auxiliary stability data and find, in particular, that in degree g there is a unique compactified Jacobian encoding slop stability, and it is induced by the tropical break divisor decomposition.

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arXiv2018

Combinatorics of compactified universal Jacobians

We use orientations on stable graphs to express the combinatorial structure of the compactified universal Jacobians in degrees g-1 and g over the moduli space of stable curves, \Mgb, and construct for them graded stratifications compatible with the one of \Mgb. In particular, for a stable curve we exhibit graded stratifications of the compactified Jacobians in terms of totally cyclic, respectively rooted, orientations on subgraphs of its dual graph.

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