1. CJM 2016 (vol 69 pp. 1274)
 Favacchio, Giuseppe; Guardo, Elena

The Minimal Free Resolution of Fat Almost Complete Intersections in $\mathbb{P}^1\times \mathbb{P}^1$
A current research theme is to compare symbolic powers of an
ideal
$I$ with the regular powers of $I$. In this paper, we focus on
the
case that $I=I_X$ is an ideal defining an almost complete
intersection (ACI) set of points $X$ in
$\mathbb{P}^1 \times \mathbb{P}^1$.
In particular,
we describe a minimal free bigraded resolution of a non
arithmetically CohenMacaulay (also non homogeneous) set $\mathcal
Z$ of fat
points whose support is an ACI, generalizing
a result of S. Cooper et al.
for homogeneous sets of triple points. We call
$\mathcal Z$ a fat ACI. We also show that its symbolic and ordinary
powers are equal, i.e,
$I_{\mathcal Z}^{(m)}=I_{\mathcal Z}^{m}$ for any $m\geq 1.$
Keywords:points in $\mathbb{P}^1\times \mathbb{P}^1$, symbolic powers, resolution, arithmetically CohenMacaulay Categories:13C40, 13F20, 13A15, 14C20, 14M05 

2. CJM 2012 (vol 65 pp. 823)
 Guardo, Elena; Harbourne, Brian; Van Tuyl, Adam

Symbolic Powers Versus Regular Powers of Ideals of General Points in $\mathbb{P}^1 \times \mathbb{P}^1$
Recent work of EinLazarsfeldSmith and HochsterHuneke
raised the problem of which symbolic powers of an ideal
are contained in a given ordinary power of the ideal.
BocciHarbourne developed methods to address this problem,
which involve asymptotic numerical characters of
symbolic powers of the ideals. Most of the work
done up to now has been done for ideals defining 0dimensional
subschemes of projective space.
Here we focus on certain subschemes given by
a union of lines in $\mathbb{P}^3$ which can also be viewed
as points in $\mathbb{P}^1 \times \mathbb{P}^1$.
We also obtain results on the
closely related problem, studied by Hochster and by LiSwanson, of
determining situations for which
each symbolic power of an ideal is an ordinary power.
Keywords:symbolic powers, multigraded, points Categories:13F20, 13A15, 14C20 

3. CJM 2008 (vol 60 pp. 391)
 Migliore, Juan C.

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 3space. 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 Glinkage, the
implementations use very different methods.
Keywords:Weak Lefschetz Property, Strong Lefschetz Property, basic double Glinkage, level, arithmetically Gorenstein, arithmetically CohenMacaulay, socle type, socle degree, Artinian reduction Categories:13D40, 13D02, 14C20, 13C40, 13C13, 14M05 

4. CJM 2005 (vol 57 pp. 724)
 Purnaprajna, B. P.

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 basepoint 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 

5. CJM 2003 (vol 55 pp. 561)
6. CJM 2002 (vol 54 pp. 554)
 Hausen, Jürgen

Equivariant Embeddings into Smooth Toric Varieties
We characterize embeddability of algebraic varieties into smooth toric
varieties and prevarieties. Our embedding results hold also in an
equivariant context and thus generalize a wellknown embedding theorem
of Sumihiro on quasiprojective $G$varieties. The main idea is to
reduce the embedding problem to the affine case. This is done by
constructing equivariant affine conoids, a tool which extends the
concept of an equivariant affine cone over a projective $G$variety to
a more general framework.
Categories:14E25, 14C20, 14L30, 14M25 

7. CJM 1998 (vol 50 pp. 1209)
 Fukuma, Yoshiaki

A lower bound for $K_X L$ of quasipolarized surfaces $(X,L)$ with nonnegative Kodaira dimension
Let $X$ be a smooth projective surface over the complex
number field and let $L$ be a nefbig divisor on $X$. Here we consider
the following conjecture; If the Kodaira dimension $\kappa(X)\geq 0$,
then $K_{X}L\geq 2q(X)4$, where $q(X)$ is the irregularity of $X$. In
this paper, we prove that this conjecture is true if (1) the case in which
$\kappa(X)=0$ or $1$, (2) the case in which $\kappa(X)=2$ and $h^{0}(L)\geq
2$, or (3) the case in which $\kappa(X)=2$, $X$ is minimal, $h^{0}(L)=1$,
and $L$ satisfies some conditions.
Keywords:Quasipolarized surface, sectional genus Category:14C20 
