Expand all Collapse all | Results 1 - 7 of 7 |
1. CJM 2010 (vol 62 pp. 1131)
Moduli Spaces of Reflexive Sheaves of Rank 2
Let $\mathcal{F}$ be a coherent rank $2$ sheaf on a scheme $Y \subset \mathbb{P}^{n}$ of
dimension at least two and let $X \subset Y$ be the zero set of a section
$\sigma \in H^0(\mathcal{F})$. In this paper, we study the relationship between the
functor that deforms the pair $(\mathcal{F},\sigma)$ and the two functors that deform
$\mathcal{F}$ on $Y$, and $X$ in $Y$, respectively. By imposing some conditions on two
forgetful maps between the functors, we prove that the scheme structure of
\emph{e.g.,} the moduli scheme ${\rm M_Y}(P)$ of stable sheaves on a threefold $Y$
at $(\mathcal{F})$, and the scheme structure at $(X)$ of the Hilbert scheme of curves
on $Y$ become closely related. Using this relationship, we get criteria for the
dimension and smoothness of $ {\rm M_{Y}}(P)$ at $(\mathcal{F})$, without assuming $
{\textrm{Ext}^2}(\mathcal{F} ,\mathcal{F} ) = 0$. For reflexive sheaves on $Y=\mathbb{P}^{3}$ whose
deficiency module $M = H_{*}^1(\mathcal{F})$ satisfies $ {_{0}\! \textrm{Ext}^2}(M ,M ) = 0 $
(\emph{e.g.,} of diameter at most 2),
we get necessary and sufficient conditions of unobstructedness that coincide
in the diameter one case. The conditions are further equivalent to the
vanishing of certain graded Betti numbers of the free graded minimal
resolution of $H_{*}^0(\mathcal{F})$. Moreover, we show that every irreducible
component of ${\rm M}_{\mathbb{P}^{3}}(P)$ containing a reflexive sheaf of diameter
one is reduced (generically smooth) and we compute its dimension. We also
determine a good lower bound for the dimension of any component of ${\rm
M}_{\mathbb{P}^{3}}(P)$ that contains a reflexive stable sheaf with ``small''
deficiency module $M$.
Keywords:moduli space, reflexive sheaf, Hilbert scheme, space curve, Buchsbaum sheaf, unobstructedness, cup product, graded Betti numbers.xdvi Categories:14C05, qqqqq14D22, 14F05, 14J10, 14H50, 14B10, 13D02, 13D07 |
2. CJM 2009 (vol 61 pp. 888)
Face Ring Multiplicity via CM-Connectivity Sequences The multiplicity conjecture of Herzog, Huneke, and Srinivasan
is verified for the face rings of the following classes of
simplicial complexes: matroid complexes, complexes of dimension
one and two,
and Gorenstein complexes of dimension at most four.
The lower bound part of this conjecture is also established for the
face rings of all doubly Cohen--Macaulay complexes whose 1-skeleton's
connectivity does not exceed the codimension plus one as well as for
all $(d-1)$-dimensional $d$-Cohen--Macaulay complexes.
The main ingredient of the proofs is a new interpretation
of the minimal shifts in the resolution of the face ring
$\field[\Delta]$ via the Cohen--Macaulay connectivity of the
skeletons of $\Delta$.
Categories:13F55, 52B05;, 13H15;, 13D02;, 05B35 |
3. CJM 2009 (vol 61 pp. 29)
The Minimal Resolution Conjecture for Points on the Cubic Surface In this paper we prove that a generalized version of the Minimal
Resolution Conjecture given by Musta\c{t}\v{a} holds for certain
general sets of points on a smooth cubic surface $X \subset
\PP^3$. The main tool used is Gorenstein liaison theory and, more
precisely, the relationship between the free resolutions of two linked schemes.
Categories:13D02, 13C40, 14M05, 14M07 |
4. CJM 2008 (vol 60 pp. 391)
The Geometry of the Weak Lefschetz Property and Level Sets of Points In a recent paper, F. Zanello showed that level Artinian algebras in 3
variables can fail to have the Weak Lefschetz Property (WLP), and can
even fail to have unimodal Hilbert function. We show that the same is
true for the Artinian reduction of reduced, level sets of points in
projective 3-space. Our main goal is to begin an understanding of how
the geometry of a set of points can prevent its Artinian reduction
from having WLP, which in itself is a very algebraic notion. More
precisely, we produce level sets of points whose Artinian reductions
have socle types 3 and 4 and arbitrary socle degree $\geq 12$ (in the
worst case), but fail to have WLP. We also produce a level set of
points whose Artinian reduction fails to have unimodal Hilbert
function; our example is based on Zanello's example. Finally, we show
that a level set of points can have Artinian reduction that has WLP
but fails to have the Strong Lefschetz Property. While our
constructions are all based on basic double G-linkage, the
implementations use very different methods.
Keywords:Weak Lefschetz Property, Strong Lefschetz Property, basic double G-linkage, level, arithmetically Gorenstein, arithmetically Cohen--Macaulay, socle type, socle degree, Artinian reduction Categories:13D40, 13D02, 14C20, 13C40, 13C13, 14M05 |
5. CJM 2005 (vol 57 pp. 724)
Some Results on Surfaces of General Type In this article we prove some new results on projective normality, normal
presentation and higher syzygies for surfaces of general type, not
necessarily smooth, embedded by adjoint linear series. Some of the
corollaries of more general results include: results on property $N_p$
associated to $K_S \otimes B^{\otimes n}$ where $B$ is base-point free and
ample divisor with $B\otimes K^*$ {\it nef}, results for pluricanonical
linear systems and results giving effective bounds for adjoint linear series
associated to ample bundles. Examples in the last section show that the results
are optimal.
Categories:13D02, 14C20, 14J29 |
6. CJM 2004 (vol 56 pp. 716)
Fat Points in $\mathbb{P}^1 \times \mathbb{P}^1$ and Their Hilbert Functions We study the Hilbert functions of fat points in $\popo$.
If $Z \subseteq \popo$ is an arbitrary fat point scheme, then
it can be shown that for every $i$ and $j$ the values of the Hilbert
function $_{Z}(l,j)$ and $H_{Z}(i,l)$ eventually become constant for
$l \gg 0$. We show how to determine these eventual values
by using only the multiplicities of the points, and the
relative positions of the points in $\popo$. This enables
us to compute all but a finite number values of $H_{Z}$
without using the coordinates of points.
We also characterize the ACM fat point schemes
sing our description of the eventual behaviour. In fact,
n the case that $Z \subseteq \popo$ is ACM, then
the entire Hilbert function and its minimal free resolution
depend solely on knowing the eventual values of the Hilbert function.
Keywords:Hilbert function, points, fat points, Cohen-Macaulay, multi-projective space Categories:13D40, 13D02, 13H10, 14A15 |
7. CJM 2000 (vol 52 pp. 123)
An Algorithm for Fat Points on $\mathbf{P}^2 Let $F$ be a divisor on the blow-up $X$ of $\pr^2$ at $r$ general
points $p_1, \dots, p_r$ and let $L$ be the total transform of a
line on $\pr^2$. An approach is presented for reducing the
computation of the dimension of the cokernel of the natural map
$\mu_F \colon \Gamma \bigl( \CO_X(F) \bigr) \otimes \Gamma \bigl(
\CO_X(L) \bigr) \to \Gamma \bigl( \CO_X(F) \otimes \CO_X(L) \bigr)$
to the case that $F$ is ample. As an application, a formula for
the dimension of the cokernel of $\mu_F$ is obtained when $r = 7$,
completely solving the problem of determining the modules in
minimal free resolutions of fat point subschemes\break
$m_1 p_1 + \cdots + m_7 p_7 \subset \pr^2$. All results hold for
an arbitrary algebraically closed ground field~$k$.
Keywords:Generators, syzygies, resolution, fat points, maximal rank, plane, Weyl group Categories:13P10, 14C99, 13D02, 13H15 |