37 research outputs found
Toric ideals of simple surface singularities
In this paper, we study a class of toric ideals obtained by using some
geometric data of ADE trees which are the minimal resolution graphs of rational
surface singularities. We compute explicit Gr\"obner bases for these toric
ideals that are also minimal generating sets consisting of large number of
binomials of degree . In particular, they give rise to squarefree
initial ideals as well
The genus of projective curves on complete intersection surfaces
We bound the genus of a projective curve lying on a complete intersection
surface in terms of its degree and the degrees of the defining equations of the
surface on which it lies.Comment: 19 pages, 4 figures, v2: references adde
On the locus of curves with an odd subcanonical marked point
We present an explicit construction of a compactification of the locus of
smooth curves whose symmetric Weierstrass semigroup at a marked point is odd.
The construction is an extension of Stoehr's techniques using
Pinkham'sequivariant deformation of monomial curves by exploring syzygies. As
an application we prove the rationality of the locus for genus at most six.Comment: 20 page
High rank linear syzygies on low rank quadrics
We study the linear syzygies of a homogeneous ideal I in a polynomial ring S,
focusing on the graded betti numbers b_(i,i+1) = dim_k Tor_i(S/I, k)_(i+1). For
a variety X and divisor D with S = Sym(H^0(D)*), what conditions on D ensure
that b_(i,i+1) is nonzero? Eisenbud has shown that a decomposition D = A + B
such that A and B have at least two sections gives rise to determinantal
equations (and corresponding syzygies) in I_X; and conjectured that if I_2 is
generated by quadrics of rank at most 4, then the last nonvanishing b_(i,i+1)
is a consequence of such equations. We describe obstructions to this conjecture
and prove a variant. The obstructions arise from toric specializations of the
Rees algebra of Koszul cycles, and we give an explicit construction of toric
varieties with minimal linear syzygies of arbitrarily high rank. This gives one
answer to a question posed by Eisenbud and Koh about specializations of
syzygies.Comment: 16 pages, 3 figure
A Bijection Between the Recurrent Configurations of a Hereditary Chip-Firing Model and Spanning Trees
Hereditary chip-firing models generalize the Abelian sandpile model and the
cluster firing model to an exponential family of games induced by covers of the
vertex set. This generalization retains some desirable properties, e.g.
stabilization is independent of firings chosen and each chip-firing equivalence
class contains a unique recurrent configuration. In this paper we present an
explicit bijection between the recurrent configurations of a hereditary
chip-firing model on a graph and its spanning trees.Comment: 13 page
Permutation-equivariant quantum K-theory IX. Quantum Hirzebruch-Riemann-Roch in all genera
We introduce the most general to date version of the permutation-equivariant
quantum K-theory, and express its total descendant potential in terms of
cohomological Gromov-Witten invariants. This is the higher-genus analogue of
adelic characterization from the paper [7] by Givental-Tonita, and is based on
the application of Kawasaki's Riemann-Roch formula to moduli spaces of stable
maps
Notes on toric varieties
These notes survey some basic results in toric varieties over a field with
examples and applications.
A computer algebra package (written by the second author) is described which
deals with both affine and projective toric varieties in any number of
dimensions (written in both the software packages MAGMA and GAP). Among other
things, the package implements a desingularization procedure for affine toric
varieties, constructs some error-correcting codes associated with toric
varieties, and computes the Riemann-Roch space of a divisor on a toric variety.Comment: 71 pages, 35 figures, inde
Tutte Short Exact Sequences of Graphs
We associate two modules, the -parking critical module and the toppling
critical module, to an undirected connected graph . We establish a
Tutte-like short exact sequence relating the modules associated to , an edge
contraction and edge deletion ( is a non-bridge). As
applications of these short exact sequences, we relate the vanishing of certain
combinatorial invariants (the number of acyclic orientations on connected
partition graphs satisfying a unique sink property) of to the equality of
corresponding invariants of and . We also obtain a short
proof of a theorem of Merino that the critical polynomial of a graph is an
evaluation of its Tutte polynomial.Comment: 40 pages, 3 figure
Divisors on graphs, binomial and monomial ideals, and cellular resolutions
We study various binomial and monomial ideals arising in the theory of divisors, orientations, and matroids on graphs. We use ideas from potential theory on graphs and from the theory of Delaunay decompositions for lattices to describe their minimal polyhedral cellular free resolutions. We show that the resolutions of all these ideals are closely related and that their Z-graded Betti tables coincide. As corollaries, we give conceptual proofs of conjectures and questions posed by Postnikov and Shapiro, by Manjunath and Sturmfels, and by Perkinson, Perlman, and Wilmes. Various other results related to the theory of chip-firing games on graphs also follow from our general techniques and results
Eigenschemes and the Jordan canonical form
We study the eigenscheme of a matrix which encodes information about the
eigenvectors and generalized eigenvectors of a square matrix. The two main
results in this paper are this decomposition encodes the numeric data of the
Jordan canonical form of the matrix. We also describe how the eigenscheme can
be interpreted as the zero locus of a global section of the tangent bundle on
projective space. This interpretation allows one to see eigenvectors and
generalized eigenvectors of matrices from an alternative viewpoint.Comment: 23 pages; v3: minor corrections, final version as appeared in Linear
Algebra App