Monomial ideals and points in projective space

Extremal Point Sets and Gorenstein Ideals

“…The proof relies heavily on the work done in [8]. The reader is advised to consult that paper (especially Example 2.4) to understand what motivates this proof.…”

Artinian level modules of embedding dimension two

“…, f α r ), then S/I is a lifting of R/I . Geramita et al proved the following (see [4,Theorem 2.2]). …”

“…Furthermore, regarded as a sheaf on P n k , N is isomorphic to the structure sheaf of the reduced scheme defined by these points. Geramita et al showed in [4] that if R/I is an artinian algebra, where I is a monomial ideal, then R/I can be lifted to a reduced set of points in P n k . We will now explain this result in more detail.…”

“…Remark 5.3. The setting used by Geramita et al in [4] is a little more general allowing the field k to be finite with some restriction on its cardinality and the points corresponding to the monomials to be chosen in different ways.…”

“…In Section 5 we use a result of Geramita et al [4] about liftings of monomial ideals. We prove that an artinian quotient of monomial ideals may be lifted to an ideal in the homogeneous coordinate ring of a certain set of reduced points such that the lifted ideal and the homogeneous coordinate ring coincide in high enough degrees.…”