arXiv subjects

Nathan Pflueger

Publications and source records attributed to Nathan Pflueger.

22 records · Page 2Linked to original sources

arXiv2016

Genera of Brill-Noether curves and staircase paths in Young tableaux

In this paper, we compute the genus of the variety of linear series of rank $r$ and degree $d$ on a general curve of genus $g$, with ramification at least $α$ and $β$ at two given points, when that variety is 1-dimensional. Our proof uses degenerations and limit linear series along with an analysis of random staircase paths in Young tableaux, and produces an explicit scheme-theoretic description of the limit linear series of fixed rank and degree on a generic chain of elliptic curves when that scheme is itself a curve.

math.AG↗

arXiv2015

Bitangents of tropical plane quartic curves

We study smooth tropical plane quartic curves and show that they satisfy certain properties analogous to (but also different from) smooth plane quartics in algebraic geometry. For example, we show that every such curve admits either infinitely many or exactly 7 bitangent lines. We also prove that a smooth tropical plane quartic curve cannot be hyperelliptic.

math.AG↗

arXiv2013

On linear series with negative Brill-Noether number

Brill-Noether theory studies the existence and deformations of curves in projective spaces; its basic object of study is $\mathcal{W}^r_{d,g}$, the moduli space of smooth genus $g$ curves with a choice of degree $d$ line bundle having at least $(r+1)$ independent global sections. The Brill-Noether theorem asserts that the map $\mathcal{W}^r_{d,g} \rightarrow \mathcal{M}_g$ is surjective with general fiber dimension given by the number $ρ= g - (r+1)(g-d+r)$, under the hypothesis that $0 \leq ρ\leq g$. One may naturally conjecture that for $ρ< 0$, this map is generically finite onto a subvariety of codimension $-ρ$ in $\mathcal{M}_g$. This conjecture fails in general, but seemingly only when $-ρ$ is large compared to $g$. This paper proves that this conjecture does hold for at least one irreducible component of $\mathcal{W}^r_{d,g}$, under the hypothesis that $0 < -ρ\leq \frac{r}{r+2} g - 3r+3$. We conjecture that this result should hold for all $0 < -ρ\leq g + C$ for some constant $C$, and we give a purely combinatorial conjecture that would imply this stronger result.

math.AG↗

arXiv2011

Graph reductions, binary rank, and pivots in gene assembly

We describe a graph reduction operation, generalizing three graph reduction operations related to gene assembly in ciliates. The graph formalization of gene assembly considers three reduction rules, called the positive rule, double rule, and negative rule, each of which removes one or two vertices from a graph. The graph reductions we define consist precisely of all compositions of these rules. We study graph reductions in terms of the adjacency matrix of a graph over the finite field with two elements, and show that they are path invariant, in the sense that the result of a sequence of graph reductions depends only on the vertices removed. The binary rank of a graph is the rank of its adjacency matrix over the finite field with two elements. We show that the binary rank of a graph determines how many times the negative rule is applied in any sequence of positive, double, and negative rules reducing the graph to the empty graph, resolving two open problems posed by Harju, Li, and Petre. We also demonstrate the close relation between graph reductions and the matrix pivot operation, both of which can be studied in terms of the poset of subsets of vertices of a graph that can be removed by a graph reduction.

math.CO↗