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1. CJM Online first
Cremona Maps of de JonquiÃ¨res Type This paper is concerned with suitable generalizations of a plane de
JonquiÃ¨res map to higher dimensional space
$\mathbb{P}^n$ with $n\geq 3$.
For each given point of $\mathbb{P}^n$ there is a subgroup of the entire
Cremona group of dimension $n$
consisting of such maps.
One studies both geometric and group-theoretical properties of this notion.
In the case where $n=3$ one describes an explicit set of generators of
the group and gives a homological characterization
of a basic subgroup thereof.
Keywords:Cremona map, de JonquiÃ¨res map, Cremona group, minimal free resolution Categories:14E05, 13D02, 13H10, 14E07, 14M05, 14M25 |
2. 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 |
3. 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 |