22 research outputs found
Stable divisorial gonality is in NP
Divisorial gonality and stable divisorial gonality are graph parameters,
which have an origin in algebraic geometry. Divisorial gonality of a connected
graph can be defined with help of a chip firing game on . The stable
divisorial gonality of is the minimum divisorial gonality over all
subdivisions of edges of .
In this paper we prove that deciding whether a given connected graph has
stable divisorial gonality at most a given integer belongs to the class NP.
Combined with the result that (stable) divisorial gonality is NP-hard by
Gijswijt, we obtain that stable divisorial gonality is NP-complete. The proof
consist of a partial certificate that can be verified by solving an Integer
Linear Programming instance. As a corollary, we have that the number of
subdivisions needed for minimum stable divisorial gonality of a graph with
vertices is bounded by for a polynomial
Recognizing hyperelliptic graphs in polynomial time
Recently, a new set of multigraph parameters was defined, called
"gonalities". Gonality bears some similarity to treewidth, and is a relevant
graph parameter for problems in number theory and multigraph algorithms.
Multigraphs of gonality 1 are trees. We consider so-called "hyperelliptic
graphs" (multigraphs of gonality 2) and provide a safe and complete sets of
reduction rules for such multigraphs, showing that for three of the flavors of
gonality, we can recognize hyperelliptic graphs in O(n log n+m) time, where n
is the number of vertices and m the number of edges of the multigraph.Comment: 33 pages, 8 figure
Discrete and metric divisorial gonality can be different
This paper compares the divisorial gonality of a finite graph G to the divisorial gonality of the associated metric graph Γ(G,1) with unit lengths. We show that dgon(Γ(G,1)) is equal to the minimal divisorial gonality of all regular subdivisions of G, and we provide a class of graphs for which this number is strictly smaller than the divisorial gonality of G. This settles a conjecture of M. Baker [3, Conjecture 3.14] in the negative
The generic Green-Lazarsfeld secant conjecture
Generalizing the well-known Green Conjecture on syzygies of canonical curves,
Green and Lazarsfeld formulated in 1986 the Secant Conjecture predicting that a
line bundle L of sufficiently high degree on a curve has a non-linear p-syzygy
if and only if L fails to be (p+1)-very ample. Via lattice theory for special
K3 surfaces, Voisin's solution of the classical Green Conjecture and
calculations on moduli stacks of pointed curves, we prove: (1) The
Green-Lazarsfeld Secant Conjecture in various degree of generality, including
its strongest possible form in the divisorial case in the universal Jacobian.
(2) The Prym-Green Conjecture on the naturality of the resolution of a general
Prym-canonical curve of odd genus.Comment: 24 pages. Final version, to appear in Inventiones Mat