139 research outputs found
Questions of Connectedness of the Hilbert Scheme of Curves in
We review the present state of the problem, for each degree and genus
, is the Hilbert scheme of locally Cohen--Macaulay curves in
connected?Comment: 10 page
Some Examples of Gorenstein Liaison in Codimension Three
Gorenstein liaison seems to be the natural notion to generalize to higher
codimension the well-known results about liaison of varieties of codimension~2
in projective space. In this paper we study points in and
curves in in an attempt to see how far typical codimension~2
results will extend. While the results are satisfactory for small degree, we
find in each case examples where we cannot decide the outcome. These examples
are candidates for counterexamples to the hoped-for extensions of codimension~2
theorems.Comment: 29 page
Publication history of von Staudt's Geometrie der Lage
From a census of forty copies, we can distinguish three different editions of
von Staudt's Geometrie der Lage: the first of 1847 and two undated ones from
the 1870's.Comment: 4 page
Clifford Index of ACM Curves in
In this paper we review the notions of gonality and Clifford index of an
abstract curve. For a curve embedded in a projective space, we investigate the
connection between the \ci of the curve and the \gc al properties of its \emb.
In particular if is a curve of degree in , and if is a
multisecant of maximum order , then the pencil of planes through cuts
out a on . If the gonality of is equal to we say the
gonality of can be computed by multisecants. We discuss the question
whether the \go of every smooth ACM curve in can be computed by
multisecants, and we show the answer is yes in some special cases.Comment: 13 page
On Rao's Theorems and the Lazarsfeld-Rao Property
Let be an integral projective scheme satisfying the condition of
Serre and for all . We
generalize Rao's theorem by showing that biliaison equivalence classes of
codimension two subschemes without embedded components are in one-to-one
correspondence with pseudo-isomorphism classes of coherent sheaves on
satisfying certain depth conditions.
We give a new proof and generalization of Strano's strengthening of the
Lazarsfeld--Rao property, showing that if a codimension two subscheme is not
minimal in its biliaison class, then it admits a strictly descending elementary
biliaison.
For a three-dimensional arithmetically Gorenstein scheme , we show that
biliaison equivalence classes of curves are in one-to-one correspondence with
triples , up to shift, where is the Rao module, is a
maximal Cohen--Macaulay module on the homogeneous coordinate ring of , and
is a surjective map of the duals.Comment: 17 page
Experiments with Gorenstein Liaison
We give some experimental data of Gorenstein liaison, working with points in
and curves in , to see how far the familiar
situation of liaison, biliaison, and Rao modules of curves in
will extend to subvarieties of codimension 3 in higher .Comment: 13 page
Geometry of arithmetically Gorenstein curves in
We characterize the postulation character of arithmetically Gorenstein curves
in . We give conditions under which the curve can be realized in
the form on some ACM surface. Finally, we strengthen a theorem of
Watanabe by showing that any general arithmetically Gorenstein curve in
can be obtained from a line by a series of ascending
complete-intersection biliaisons.Comment: 15 page
Liaison with Cohen-Macaulay Modules
We describe some recent work concerning Gorenstein liaison of codimension two
subschemes of a projective variety. Applications make use of the algebraic
theory of maximal Cohen-Macaulay modules, which we review in an Appendix.Comment: 16 page
Curves in the double plane
We study locally Cohen-Macaulay curves in projective three-space which are
contained in a double plane 2H, thus completing the classification of curves
lying on surfaces of degree two. We describe the irreducible components of the
Hilbert schemes of locally Cohen-Macaulay curves in 2H of given degree and
arithmetic genus. We show that these Hilbert schemes are connected. We also
discuss the Rao modules of these curves, and liaison and biliaison equivalence
classes.Comment: 20 page
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