TY - JOUR
T1 - Hilbert schemes of 8 points
AU - Cartwright, Dustin A.
AU - Erman, Daniel
AU - Velasco, Mauricio
AU - Viray, Bianca
PY - 2009
Y1 - 2009
N2 - The Hilbert scheme Hdn of n points in Ad contains an irreducible component Rdn which genetically represents n distinct points in Ad. We show that when n is at most 8, the Hilbert scheme Hdn is reducible if and only if n = 8 and d ≥ 4. In the simplest case of reducibility, the component R 48 ⊂ H48 is defined by a single explicit equation, which serves as a criterion for deciding whether a given ideal is a limit of distinct points. To understand the components of the Hilbert scheme, we study the closed subschemes of Hdn which parametrize those ideals which are homogeneous and have a fixed Hilbert function. These subschemes are a special case of multi-graded Hilbert schemes, and we describe their components when the colength is at most 8. In particular, we show that the scheme corresponding to the Hilbert function (1, 3, 2, 1) is the minimal reducible example.
AB - The Hilbert scheme Hdn of n points in Ad contains an irreducible component Rdn which genetically represents n distinct points in Ad. We show that when n is at most 8, the Hilbert scheme Hdn is reducible if and only if n = 8 and d ≥ 4. In the simplest case of reducibility, the component R 48 ⊂ H48 is defined by a single explicit equation, which serves as a criterion for deciding whether a given ideal is a limit of distinct points. To understand the components of the Hilbert scheme, we study the closed subschemes of Hdn which parametrize those ideals which are homogeneous and have a fixed Hilbert function. These subschemes are a special case of multi-graded Hilbert schemes, and we describe their components when the colength is at most 8. In particular, we show that the scheme corresponding to the Hilbert function (1, 3, 2, 1) is the minimal reducible example.
KW - Hilbert scheme
KW - Smoothable
KW - Zero-dimensional ideal
UR - http://www.scopus.com/inward/record.url?scp=77954016930&partnerID=8YFLogxK
U2 - 10.2140/ant.2009.3.763
DO - 10.2140/ant.2009.3.763
M3 - Artículo
AN - SCOPUS:77954016930
SN - 1937-0652
VL - 3
SP - 763
EP - 795
JO - Algebra and Number Theory
JF - Algebra and Number Theory
IS - 7
ER -