1,184 research outputs found
Bound on the projective dimension of three cubics
We show that given any polynomial ring R over a field, and any ideal J in R
which is generated by three cubic forms, the projective dimension of R/J is at
most 36. We also settle the question whether ideals generated by three cubic
forms can have projective dimension greater than 4, by constructing one with
projective dimension equal to 5.Comment: to appear in Journal of Symbolic Computatio
On the projective dimension and the unmixed part of three cubics
Let be a polynomial ring over a field in an unspecified number of
variables. We prove that if is an ideal generated by three cubic
forms, and the unmixed part of contains a quadric, then the projective
dimension of is at most 4. To this end, we show that if is
a three-generated ideal of height two and an ideal linked to the
unmixed part of , then the projective dimension of is bounded above by
the projective dimension of plus one.Comment: 23 pages; to appear in Journal of Algebr
Projective varieties with many degenerate subvarieties
We study the problem of classifying the irreducible projective varieties
of dimension in which contain an algebraic family \Cal F
of dimension () of subvarieties of dimension , each one
contained in a . We prove that one of the following happens:
(i) there exists an integer , such that is contained in a
variety of dimension at most containing a family of dimension
of subvarieties of dimension , each one contained in a linear space of
dimension ; (ii) The degree of is bounded by a function of and
(in this case is called of isolated type). Successively we study some
special cases; in particular we give a complete classification of surfaces in
containing a family of dimension of curves of .Comment: 19 pages, AMS-TeX 2.
Hilbert's fourteenth problem over finite fields, and a conjecture on the cone of curves
We give examples over arbitrary fields of rings of invariants that are not
finitely generated. The group involved can be as small as three copies of the
additive group, as in Mukai's examples over the complex numbers. The failure of
finite generation comes from certain elliptic fibrations or abelian surface
fibrations having positive Mordell-Weil rank.
Our work suggests a generalization of the Morrison-Kawamata cone conjecture
from Calabi-Yau varieties to klt Calabi-Yau pairs. We prove the conjecture in
dimension 2 in the case of minimal rational elliptic surfaces.Comment: 26 pages. To appear in Compositio Mathematic
On the classification of OADP varieties
The main purpose of this paper is to show that OADP varieties stand at an
important crossroad of various main streets in different disciplines like
projective geometry, birational geometry and algebra. This is a good reason for
studying and classifying them. Main specific results are: (a) the
classification of all OADP surfaces (regardless to their smoothness); (b) the
classification of a relevant class of normal OADP varieties of any dimension,
which includes interesting examples like lagrangian grassmannians. Following
[PR], the equivalence of the classification in (b) with the one of
quadro-quadric Cremona transformations and of complex, unitary, cubic Jordan
algebras are explained.Comment: 13 pages. Dedicated to Fabrizio Catanese on the occasion of his 60th
birthday. To appear in a special issue of Science in China Series A:
Mathematic
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