Generic Polarised Varieties
Recall from Section Key Warning and Disclaimer that, despite some ambiguity, we regard the following as being equivalent: polarised varieties \(X,A\); schemes in wps \(X\subset\P^N(w_0,\dots,w_N)\) where \(A\) is a degree 1 hyperplane section; and data about the Hilbert series of the graded ring \(R(X,A)\). Thus we constantly refer to a polarised variety \(X\), and we expect to be able to retrieve its Hilbert series, its dimension, the codimension of its embedding, and other such data.
With one exception, the intrinsics described in this section can be applied to all polarised varieties. The exception is the following little-used intrinsic that creates a polarised variety that is not of a specific type.
- PolarisedVariety(d, W, n): RngIntElt, SeqEnum, RngUPolElt -> GRSch
The polarised variety of dimension \(d\), with weights given by the sequence \(W\) of positive integers and with Hilbert numerator the univariate polynomial \(n\).
Accessing the Data
- RawBasket(X): GRSch -> SeqEnum
The basket of singularities of the polarised variety \(X\) in sequence format, that is, the basket is a sequence of sequences, in which a singularity \({{1}\over{r}}(a,b,c)\) is represented as a sequence \([r,a,b,c]\) of integers. (The Gorenstein surface singularity \({{1}\over{r}}(a,r-a)\) admits further abbreviation to \([r,a]\).) Notice that the local polarisation \(n\) is not included in this raw basket data; its default value \(n=-1\) is assumed.
- HilbertNumerator(X): GRSch -> RngUPolElt
- Numerator(X): GRSch -> RngUPolElt
The numerator \(f(t)\) of the Hilbert series \(P(t)\) of the polarised variety \(X\) when expressed as a rational function \(P=f(t)/ \&* [1-t^w : w \in W]\) where the product in the denominator is taken over \(W\), the sequence of weights of \(X\).
- NoetherWeights(X): GRSch -> SeqEnum
The weights corresponding to a Noether normalisation of the polarised variety \(X\). In other words, these are the weights of polynomials in the graded ring of \(X\) that generate a polynomial subring of maximal dimension.
- NoetherNumerator(X): GRSch -> RngUPolElt
The numerator \(n(t)\) of the Hilbert series \(P(t)\) of the polarised variety \(X\) when expressed as a rational function \(P=n(t)/ \&* [1-t^w : w \in N ]\), where the product in the denominator is taken over \(N\), the sequence of Noether weights of \(X\).
- NoetherNormalisation(X): GRSch -> Tup
Given a polarised variety \(X\) return a pair, the first term of which is the sequence of Noether weights, the second the corresponding numerator.
- HilbertSeries(X): GRSch -> FldFunRatUElt
The Hilbert series of the polarised variety \(X\) expressed as a rational function.
- InitialCoefficients(X): GRSch -> SeqEnum
The coefficients of the Hilbert series of the polarised variety \(X\) expressed as a power series. The number of coefficients returned is equal to the precision of the power series ring in which the Hilbert series was expanded.
- ApparentCodimension(X): GRSch -> RngIntElt
- ApparentEquationDegrees(X): GRSch -> RngIntElt
- ApparentSyzygyDegrees(X): GRSch -> RngIntElt
- BettiNumbers(X): GRSch -> RngIntElt
If \(n(t)\) is the Hilbert numerator of \(X\) and is of the form
\[n = 1 - \sum_{i=1}^{N_0} a_{0,i} t^i + \sum_{k=N_0+1}^{N_1} a_{1,i} t^i - \dots +(-1)^{k-1} \sum_{i=N_{k-2}+1}^{N_{k-1}} a_{k-1,i} t^i +(-1)^k t^{N_k}\]then the apparent codimension of \(X\) is \(k\), the apparent equation degrees are given by those \(i\) for which \(a_{0,i}\) is nonzero (with \(a_{0,i}\) equations of that degree \(i\)) and the apparent syzygy degrees are those integers \(i\) for which \(a_{1,i}\) is nonzero. The Betti numbers are a sequence with first element the sum of all \(a_{0,i}\), second element the sum of all \(a_{1,i}\), and so on until \(a_{k-1,i}\).
Generic Creation, Checking, Changing
Procedural versions of intrinsic functions modify polarised varieties at the generic level because they preserve any subtypes; functional versions exist for special types of polarised variety but not in general.
- X eq Y: GRSch, GRSch -> BoolElt
Return
trueif and only if the polarised varieties \(X\) and \(Y\) have the same dimension, weights, basket, and Hilbert numerator. In particular, these conditions imply that \(X\) and \(Y\) have the same Hilbert series.
- CheckCodimension(X): GRSch -> BoolElt
Return
trueif and only if the codimension of \(X\) is equal to the apparent codimension of \(X\) determined by its Hilbert numerator.
- FirstWeights(X): GRSch -> SeqEnum
These are weights assigned to the polarised variety \(X\) during its construction that carry some relevance; if no such weights were assigned, the usual weights of \(X\) will be returned.
- IncludeWeight(~X, w): GRSch, RngIntElt
Include the positive integer \(w\) among the weights of \(X\), adjusting all other data associated to the embedding of \(X\) as required.