Abstract view
The Minimal Free Resolution of Fat Almost Complete Intersections in $\mathbb{P}^1\times \mathbb{P}^1$


Published:20161223
Printed: Dec 2017
Giuseppe Favacchio,
Dipartimento di Matematica e Informatica , Viale A. Doria, 6  95100  Catania, Italy
Elena Guardo,
Dipartimento di Matematica e Informatica , Viale A. Doria, 6  95100  Catania, Italy
Abstract
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 $m\geq 1.$
MSC Classifications: 
13C40, 13F20, 13A15, 14C20, 14M05 show english descriptions
Linkage, complete intersections and determinantal ideals [See also 14M06, 14M10, 14M12] Polynomial rings and ideals; rings of integervalued polynomials [See also 11C08, 13B25] Ideals; multiplicative ideal theory Divisors, linear systems, invertible sheaves Varieties defined by ring conditions (factorial, CohenMacaulay, seminormal) [See also 13F15, 13F45, 13H10]
13C40  Linkage, complete intersections and determinantal ideals [See also 14M06, 14M10, 14M12] 13F20  Polynomial rings and ideals; rings of integervalued polynomials [See also 11C08, 13B25] 13A15  Ideals; multiplicative ideal theory 14C20  Divisors, linear systems, invertible sheaves 14M05  Varieties defined by ring conditions (factorial, CohenMacaulay, seminormal) [See also 13F15, 13F45, 13H10]
