86 research outputs found
Horrocks splitting on Segre-Veronese varieties
We prove an analogue of Horrocks' splitting theorem for Segre-Veronese
varieties building upon the theory of Tate resolutions on products of
projective spaces
Geometry and Algebra of prime Fano 3-folds of genus 12
The connection between these Fano 3-folds and plane quartic curves is
explained.Comment: LaTex,19 page
Matrix factorizations and families of curves of genus 15
In this note, we explain how certain matrix factorizations on cubic
threefolds lead to families of curves of genus 15 and degree 16 in P^4. We
prove that the moduli space M={(C,L) | C a curve of genus 15, L a line bundle
on C of degree 16 with 5 sections} is birational to a certain space of matrix
factorizations of cubics, and that M is uniruled. Our attempt to prove the
unirationality of this space failed with our methods. Instead one can interpret
our findings as evidence for the conjecture that the basis of the maximal
rational connected fibration of M has a three dimensional base.Comment: 26 page
Unirationality of the Hurwitz space H_{9,8}
In this paper we prove that the Hurwitz space H_{9,8}, which parameterizes
8-sheeted covers of P^1 by curves of genus 9, is unirational. Our construction
leads to an explicit Macaulay2 code, which will randomly produce a nodal curve
of degree 8 of geometric genus 9 with 12 double points and together with a
pencil of degree 8
Sheaf Cohomolog and Free Resolutions over Exterior Algebras
In this paper we derive an explicit version of the
Bernstein-Gel'fand-Gel'fand (BGG) correspondence between bounded complexes of
coherent sheaves on projective space and minimal doubly infinite free
resolutions over its ``Koszul dual'' exterior algebra. This leads to an
efficient method for machine computation of the cohomology of sheaves. Among
the facts about the BGG correspondence that we derive is that taking homology
of a complex of sheaves corresponds to taking the ``linear part'' of a
resolution over the exterior algebra.
Using these results we give a constructive proof of the existence of a
Beilinson monad for a sheaf on projective space. The explicitness of our
version allows us to to prove two conjectures about the morphisms in the monad.
Along the way we prove a number of results about minimal free resolutions
over an exterior algebra. For example, we show that such resolutions are
eventually dominated by their "linear parts" in the sense that erasing all
terms of degree in the complex yields a new complex which is eventually
exact.Comment: 29 page
Betti Numbers of Graded Modules and Cohomology of Vector Bundles
Mats Boij and Jonas Soederberg (math.AC/0611081) have conjectured that the
Betti table of a Cohen-Macaulay module over a polynomial ring can be decomposed
in a certain way as a positive linear combination of Betti tables of modules
with pure resolutions. We prove, over any field, a strengthened form of their
conjecture. Applications include a proof of the Multiplicity Conjecture of
Huneke and Srinivasan and a proof of the convexity of a fan naturally
associated to the Young lattice.
We also characterize the rational cone of all cohomology tables of vector
bundles on projective spaces in terms of the cohomology tables of
"supernatural" bundles. This characterization is dual, in a certain sense, to
our characterization of Betti tables.Comment: This version incorporates many corrections and many expository
improvements of the original. It is to appear in the Journal of the American
Mathematical Societ
An explicit matrix factorization of cubic hypersurfaces of small dimension
In this paper, we compute an explicit matrix factorization of a rank 9 Ulrich
sheaf on a general cubic hypersurface of dimension at most 7, whose existence
was proved by Manivel. Instead of using invariant theory, we use Shamash's
construction with a cone over the spinor variety. We also describe an
algebro-geometric interpretation of our matrix factorization which connects the
spinor tenfold and the Cartan cubic.Comment: 18 pages, with Macaulay2 scripts. Fixed some typos, and updated a few
reference
Canonical projections of irregular algebraic surfaces
A good canonical projection of a surface of general type is a morphism to
the 3-dimensional projective space P^3 given by 4 sections of the canonical
line bundle. To such a projection one associates the direct image sheaf F of
the structure sheaf of S, which is shown to admit a certain length 1
symmetrical locally free resolution. The structure theorem gives necessary and
sufficient conditions for such a length 1 locally free symmetrical resolution
of a sheaf F on P^3 in order that yield a canonically
projected surface of general type. The result was found in 1983 by the first
author in the regular case, and the main ingredient here for the irregular (=
non Cohen-Macaulay) case is to replace the use of Hilbert resolutions with the
use of Beilinson's monads. Most of the paper is devoted then to the analysis of
irregular canonical surfaces with p_g=4 and low degree (the analysis in the
regular case had been initiated by F. Enriques). For q=1 we classify completely
the (irreducible) moduli space for the minimal value K^2=12. We also study
other moduli spaces, q=2, K^2=18, and q=3, K^2=12. The last family is provided
by the divisors in a (1,1,2) polarization on some Abelian 3-fold. A remarkable
subfamily is given by the pull backs, under a 2-isogeny, of the theta divisor
of a p.p.Abelian 3-fold: for those surfaces the canonical map is a double cover
of a canonical surface (i.e., such that its canonical map is birational). The
canonical image is a sextic surface with a plane cubic as double curve, and
moreover with an even set of 32 nodes as remaining singular locus. Our method
allows to write the equations of these transcendental objects, but not yet to
construct new ones by computer algebra.Comment: 35 pages. To appear in a volume in memory of Paolo Francia, De
Gruyter Verla
Varieties of sums of powers
The variety of sums of powers of a homogeneous polynomial of degree d in n
variables is defined and investigated in some examples, old and new. These
varieties are studied via apolarity and syzygies. Classical results of
Sylvester (1851), Hilbert (1888), Dixon and Stuart (1906) and some more recent
results of Mukai (1992) are presented together with new results for the cases
(n,d)=(3,8), (4,2), (5,3). In the last case the variety of sums of 8 powers of
a general cubic form is a Fano 5-fold of index 1 and degree 660.Comment: Final published version (Crelle
Resultants and Chow forms via Exterior Syzygies
Given a sheaf on a projective space P^n we define a sequence of canonical and
easily computable Chow complexes on the Grassmannians of planes in P^n,
generalizing the Beilinson monad on P^n. If the sheaf has dimension k, then the
Chow form of the associated k-cycle is the determinant of the Chow complex on
the Grassmannian of planes of codimension k+1. Using the theory of vector
bundles and the canonical nature of the complexes we are able to give explicit
determinantal and Pfaffian formulas for resultants in some cases where no
polynomial formulas were known. For example, the Horrocks-Mumford bundle gives
rise to a polynomial formula for the resultant of five homogeneous forms of
degree eight in five variables.Comment: 38 page
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