A partial lattice P is ideal-projective, with respect to a class C of lattices, if for every K ∈ C and every homomorphism φ of partial lattices from P to the ideal lattice of K, there are arbitrarily large choice functions f : P → K for φ that are also homomorphisms of partial lattices. This extends the traditional concept of (sharp) transferability of a lattice with respect to C. We prove the ...

We introduce the concept of multiplication matrices for ideals of projective dimension zero. We discuss various applications and in particular, we give a new algorithm to compute the variety of an ideal of projective dimension zero.

We show that given any polynomial ring R over a field and any ideal J ⊂ R which is generated by three cubic forms, the projective dimension of R/J is at most 36. We also settle the question of whether ideals generated by three cubic forms can have projective dimension greater than four, by constructing one with projective dimension equal to five.

We discuss an algorithm computing the push-forward to projective space of several classes associated to a (possibly singular, reducible, nonreduced) projective scheme. For example, the algorithm yields the topological Euler characteristic of the support of a projective scheme S, given the homogeneous ideal of S. The algorithm has been implemented in Macaulay2.

We show that given any polynomial ring R over a field and any ideal J ⊂ 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 four, by constructing one with projective dimension equal to five.

For an ideal I in a polynomial ring over a field, a monomial support of I is the set of monomials that appear as terms in a set of minimal generators of I. Craig Huneke asked whether the size of a monomial support is a bound for the projective dimension of the ideal. We construct an example to show that, if the number of variables and the degrees of the generators are unspecified, the projectiv...

We define a power series associated with a homogeneous ideal in a polynomial ring, encoding information on the Segre classes defined by extensions of the ideal in projective spaces of arbitrarily high dimension. We prove that this power series is rational, with poles corresponding to generators of the ideal, and with numerator of bounded degree and with nonnegative coefficients. We also prove t...

This report gives an overview of the main ideas in Chapter 9 of the studied book, about the dimension of a variety. After recalling some definitions and basic properties about projective varieties, homogeneous ideals and Gröbner bases, we will begin our study of the dimension of a variety with the observation of a geometric way to define the dimension of a variety defined by a monomial ideal. T...