doc ///
Key
VirtualResolutions
Headline
a package for computing virtual resolutions
Description
Text
While graded minimal free resolutions are useful for studying quasicoherent
sheaves over projective space, when working over a product of projective spaces or, more generally,
over smooth projective toric varieties, graded minimal free resolutions over the Cox ring
seem too restricted by algebraic structure that is in some sense unimportant geometrically. By allowing
a limited amount of homology, virtual resolutions offer a more flexible alternative for
studying toric subvarieties when compared to minimal graded free resolutions.
Introduced by Berkesch, Erman, and Smith in {\em Virtual resolutions for a product of projective spaces}
(see [BES20, @arXiv "1703.07631"@]) if $X$ is a smooth toric variety, $S$ is the Cox ring of $X$
graded by the Picard group of $X$, and $B\subset S$ is the irrelevant ideal of $X$, then
a virtual resolution of a graded $S$-module $M$ is a complex of graded free $S$-modules, which
sheafifies to a resolution of the associated sheaf of $M$.
This package provides tools for constructing and studying virtual resolutions for products of projective spaces.
In particular, it implements a number of the methods for constructing virtual resolutions for products of projective
spaces as introduced by Berkesch, Erman, and Smith. This package also contains methods for constructing curves in
$\PP^1\times\PP^2$, as these are a natural source for interesting virtual resolutions.
As a running example, consider three points $([1:1],[1:4])$, $([1:2],[1:5])$, and $([1:3],[1:6])$
in $\PP^1 \times \PP^1$.
Example
X = toricProjectiveSpace(1)**toricProjectiveSpace(1);
S = ring X;
B = ideal X;
J = saturate(intersect(
ideal(x_1 - x_0, x_3 - 4*x_2),
ideal(x_1 - 2*x_0, x_3 - 5*x_2),
ideal(x_1 - 3*x_0, x_3 - 6*x_2)), B);
minres = res J;
multigraded betti minres
Text
As described in Algorithm 3.4 of Berkesch, Erman, and Smith's
paper, one may construct a virtual resolution of a module from its graded minimal free resolution and
an element of the multigraded Castelnuovo-Mumford regularity of the module. (See Maclagan and Smith's paper
{\em Multigraded Castelnuovo-Mumford Regularity} (see [MS04, @arXiv "math/0305214"@]) for the definition of multigraded regularity.)
Building on the @TO TateOnProducts@ and @TO LinearTruncations@ packages, this package contains a function allowing one
to compute the minimal elements of the multigraded Castelnuovo-Mumford regularity of a $B$-saturated module.
Continuing the example from above, we see that $(2,0)$ is an element of the multigraded
regularity of $S/J$. From this we can compute a virtual resolution of $S/J$.
Example
multigradedRegularity(X, J)
vres = virtualOfPair(J, {{3,1}})
multigraded betti vres
Text
Notice that this virtual resolution of $S/J$ is much shorter and thinner than the graded minimal
free resolution of $S/J$. This is a common theme: virtual resolutions tend to be much
shorter and less wide than graded minimal free resolutions over the Cox ring, but they still
preserve geometric information about $S/J$.
In addition to the functions highlighted above, the @TT "VirtualResolutions"@ package contains
a number of other tools for constructing and studying virtual resolutions. In particular,
there are functions to construct virtual resolutions for zero dimensional subschemes, to
check whether a complex is a virtual resolution, and to construct curves in $\PP^1\times\PP^2$.
References
@UL {
{"[BES20]: Berkesch, Erman, and Smith, Virtual resolutions for a product of projective spaces (see ", arXiv "1703.07631", ")."},
{"[MS04]: Maclagan and Smith, Multigraded Castelnuovo-Mumford Regularity (see ", arXiv "math/0305214", ")."}
}@
Contributors
The following people have generously contributed code or worked on this package.
@UL {
{HREF("http://www.math.wisc.edu/~derman/","Daniel Erman")},
{HREF("https://mast.queensu.ca/~ggsmith/","Gregory G. Smith")},
{HREF("https://math.berkeley.edu/~lch/", "Lauren Cranton Heller")},
}@
///
doc ///
Key
isVirtual
(isVirtual,Ideal,ChainComplex)
(isVirtual,NormalToricVariety,ChainComplex)
Headline
checks whether a chain complex is a virtual resolution
Usage
isVirtual(irr,C)
isVirtual(X,C)
Inputs
irr:Ideal
irrelevant ideal of the ring
X:NormalToricVariety
normal toric variety
C:ChainComplex
chain complex we want to check if is a virtual resolution
Outputs
:Boolean
true if {\tt C}is a virtual resolution of I
false if not
Description
Text
Given the irrelevant ideal irr of a NormalToricVariety and a chain complex C, isVirtual returns true if
{\tt C} is a virtual resolution of some module. If not, it returns false. This is done by checking that the
higher homology groups of {\tt C}are supported on the irrelevant ideal.
If @TO "debugLevel"@ is larger than zero, the homological degree where isVirtual fails is printed.
Example
R = ZZ/101[s,t];
isVirtual(ideal(s,t),res ideal(t))
Text
Continuing our running example of three points $([1:1],[1:4])$, $([1:2],[1:5])$, and $([1:3],[1:6])$
in $\PP^1 \times \PP^1$, we can check whether the virtual complex we compute below and
in other places is in fact virtual.
Example
Y = toricProjectiveSpace(1)**toricProjectiveSpace(1);
S = ring Y;
B = ideal Y;
J = saturate(intersect(
ideal(x_1 - x_0, x_3 - 4*x_2),
ideal(x_1 - 2*x_0, x_3 - 5*x_2),
ideal(x_1 - 3*x_0, x_3 - 6*x_2)), B);
minres = res J;
vres = virtualOfPair(J,{{3,1}});
isVirtual(B,vres)
Text
Finally, we can also use the @TT "Determinantal"@ strategy, which implements Theorem 1.3 of [Loper, @arXiv "1904.05994"@].
Example
isVirtual(B,vres,Strategy=>Determinantal)
///
doc ///
Key
[isVirtual, Strategy]
Headline
changes strategy from computing homology to computing minors of boundary maps
Description
Text
If Strategy is set to @TT "Determinantal"@, isVirtual will check whether the given chain complex
is a virtual resolution by checking the depth of the saturation of the ideals of maximal rank
from the boundary maps. See Theorem 1.3 of [Loper, @arXiv "1904.05994"@].
SeeAlso
isVirtual
///
doc ///
Key
idealSheafGens
(idealSheafGens,ZZ,Ideal,Ideal)
(idealSheafGens,ZZ,Ideal,NormalToricVariety)
Headline
creates a list of subsets of the minimal generators that generate a given ideal up to saturation
Usage
idealSheafGens(n,I,irr)
idealSheafGens(n,I,X)
Inputs
I:Ideal
n:ZZ
size of subset of minimal generators of {\tt I} that may generate {\tt I} up to saturation with {\tt irr}
irr:Ideal
irrelevant ideal
X:NormalToricVariety
normal toric variety whose Cox ring contains {\tt I}
Outputs
:List
all ideals generated by subsets of size {\tt n} of generators of {\tt I} that generate {\tt I} up to saturation with {\tt irr}
Description
Text
Given an ideal {\tt I}, integer {\tt n}, and irrelevant ideal {\tt irr}, @TT "idealSheafGens"@ searches through
all {\tt n}-subsets of the generators of {\tt I}. If a subset generates the same {\tt irr}-saturated ideal as the
{\tt irr}-saturation of {\tt I}, then the ideal generated by that subset is added to a list.
After running through all subsets, the list is returned.
Example
R = ZZ/101[x_0,x_1,x_2,x_3,x_4,Degrees=>{2:{1,0},3:{0,1}}];
B = intersect(ideal(x_0,x_1),ideal(x_2,x_3,x_4));
I = ideal(x_0^2*x_2^2+x_1^2*x_3^2+x_0*x_1*x_4^2, x_0^3*x_4+x_1^3*(x_2+x_3));
idealSheafGens(2,I,B)
///
doc ///
Key
GeneralElements
[idealSheafGens, GeneralElements]
Headline
combines generators of same degree into a general linear combination
Description
Text
If @TT "GeneralElements"@ is set to true, @TO "idealSheafGens"@ will replace all generators of {\tt I} of the same degree with
a new generator of the that degree which is a general linear combination of those generators, then run @TT "idealSheafGens"@ on the new ideal.
SeeAlso
idealSheafGens
///
doc ///
Key
randomRationalCurve
(randomRationalCurve,ZZ,ZZ,Ring)
(randomRationalCurve,ZZ,ZZ)
Headline
creates the ideal of a random rational curve of degree (d,e) in P^1xP^2
Usage
randomRationalCurve(d,e,F)
randomRationalCurve(d,e)
Inputs
d:ZZ
degree of curve on the $\PP^1$ factor of $\PP^1\times\PP^2$
e:ZZ
degree of curve on the $\PP^2$ factor of $\PP^1\times\PP^2$
F:Ring
base ring
Outputs
:Ideal
defining random rational curve in $\PP^1\times\PP^2$ of degree {\tt (d,e)} over {\tt F}.
Description
Text
Given two positive integers {\tt d,e} and a ring {\tt F}, @TT "randomRationalCurve"@ returns the ideal of a
random curve in $\PP^1\times\PP^2$ of degree {\tt (d,e)} defined over the base ring {\tt F}.
This is done by randomly generating two homogeneous polynomials of degree {\tt d} and three homogeneous
polynomials of degree three in $F[s,t]$ defining maps $\PP^1\to\PP^1$ and $\PP^1\to\PP^2$,
respectively. The graph of the product of these two maps in $\PP^1\times(\PP^1\times\PP^2)$ is computed,
from which a curve of bi-degree {\tt (d,e)} in $\PP^1\times\PP^2$ over {\tt F} is obtained by
saturating and then eliminating.
If no base ring is specified, the computations are performed over {\tt ZZ/101}.
Example
randomRationalCurve(2,3,QQ);
randomRationalCurve(2,3);
Caveat
This creates a ring $F[x_{0,0},x_{0,1},x_{1,0},x_{1,1},x_{1,2}]$ in which the resulting ideal is defined.
///
doc ///
Key
randomMonomialCurve
(randomMonomialCurve,ZZ,ZZ,Ring)
(randomMonomialCurve,ZZ,ZZ)
Headline
creates the ideal of a random monomial curve of degree (d,e) in P^1xP^2
Usage
randomMonomialCurve(d,e,F)
randomMonomialCurve(d,e)
Inputs
d:ZZ
degree of curve on the $\PP^1$ factor of $\PP^1\times\PP^2$
e:ZZ
degree of curve on the $\PP^2$ factor of $\PP^1\times\PP^2$
F:Ring
base ring
Outputs
:Ideal
defining random monomial curve in $\PP^1\times\PP^2$ of degree (d,e) over F.
Description
Text
Given two positive integers {\tt d,e} and a ring {\tt F}, randomMonomialCurve returns the ideal of a
random curve in $\PP^1\times\PP^2$ of degree {\tt (d,e)} defined over the base ring {\tt F}.
This is done by randomly generating a monomial $m$ of degree $e$ in $F[s,t]$, which is not $s^e$ or $t^e$.
This allows one to define two maps $\PP^1\to\PP^1$ and $\PP^1\to\PP^2$
given by @TT"{s^d,t^d}"@ and @TT"{s^e,m,t^e}"@, respectively. The graph of the product of these two maps
in $\PP^1\times(\PP^1\times\PP^2)$ is computed, from which a curve of bi-degree {\tt (d,e)}
in $\PP^1\times\PP^2$ over {\tt F} is obtained by saturating and then eliminating.
If no base ring is specified, the computations are performed over {\tt ZZ/101}.
Example
randomMonomialCurve(2,3,QQ);
Caveat
This creates a ring $F[x_{0,0},x_{0,1},x_{1,0},x_{1,1},x_{1,2}]$ in which the resulting ideal is defined.
///
doc ///
Key
curveFromP3toP1P2
(curveFromP3toP1P2,Ideal)
Headline
creates the ideal of a curve in P^1xP^2 from the ideal of a curve in P^3
Usage
I = curveFromP3toP1P2(J)
Inputs
J:Ideal
defining a curve in $\PP^3$.
Outputs
I:Ideal
defining a curve in $\PP^1\times\PP^2$.
Description
Text
Given an ideal {\tt J} defining a curve $C$ in $\PP^3$, @TT "curveFromP3toP1P2"@ produces the ideal of the curve
in $\PP^1\times\PP^2$ defined as follows:
consider the projections $\PP^3\to\PP^2$ and $\PP^3\to\PP^1$ from the point [0:0:0:1]
and the line [0:0:s:t], respectively. The product of these defines a map from $\PP^3$ to $\PP^1\times\PP^2$.
The curve produced by @TT "curveFromP3toP1P2"@ is the image of the input curve under this map.
This computation is done by first constructing the graph in $\PP^3\times(\PP^1\times\PP^2)$ of the product
of the two projections $\PP^3\to\PP^2$ and $\PP^3\to\PP^1$ defined above.
This graph is then intersected with $C\times(\PP^1\times\PP^2)$. A curve in $\PP^1\times\PP^2$ is then
obtained from this by saturating and then eliminating.
Note the curve in $\PP^1\times\PP^2$ will have degree and genus equal to the degree and genus of $C$ as long as $C$
does not intersect the base locus of the projection. If the option @TO [curveFromP3toP1P2, PreserveDegree]@
is set to true, @TT "curveFromP3toP1P2"@ will check whether $C$ intersects the base locus.
If it does, the function will return an error. If PreserveDegree is set to false, this check is not
performed and the output curve in $\PP^1\times\PP^2$ may have degree and genus different from $C$.
Example
R = ZZ/101[z_0,z_1,z_2,z_3];
J = ideal(z_0*z_2-z_1^2, z_1*z_3-z_2^2, z_0*z_3-z_1*z_2);
curveFromP3toP1P2(J)
Caveat
This creates a ring $F[x_{0,0},x_{0,1},x_{1,0},x_{1,1},x_{1,2}]$ in which the resulting ideal is defined.
///
doc ///
Key
PreserveDegree
[curveFromP3toP1P2, PreserveDegree]
Headline
Determines if curve is disjoint from base loci
Description
Text
When set to true, @TO "curveFromP3toP1P2"@ will check whether or not the given curve
in $\PP^3$ intersects the base locus of the projections maps used in this function.
If this option is set to true and the given curve does intersect the base locus,
an error is returned.
SeeAlso
curveFromP3toP1P2
///
doc ///
Key
randomCurveP1P2
(randomCurveP1P2,ZZ,ZZ,Ring)
(randomCurveP1P2,ZZ,ZZ)
Headline
creates the ideal of a random curve in P^1xP^2
Usage
randomCurveP1P2(d,g,F)
randomCurveP1P2(d,g)
Inputs
d:ZZ
degree of the curve.
g:ZZ
genus of the curve.
F:Ring
base ring.
Outputs
:Ideal
defining random curve $\PP^1\times\PP^2$ from a curve of degree {\tt d} and genus {\tt g} in $\PP^3$ over {\tt F}.
Description
Text
Given a positive integer {\tt d}, a non-negative integer {\tt g}, and a ring {\tt F}, @TT "randomCurveP1P2"@
produces a random curve of bi-degree {\tt (d,d)} and genus {\tt g} in $\PP^1\times\PP^2$.
This is done by using the @TO "SpaceCurves::curve"@ function from the @TO SpaceCurves@ package to first generate a random curve
of degree {\tt d} and genus {\tt g} in $\PP^1\times\PP^2$, and then applying @TO "curveFromP3toP1P2"@ to produce a curve in $\PP^1\times\PP^2$.
Since @TO "curveFromP3toP1P2"@ relies on projecting from the point $[0:0:0:1]$ and the line $[0:0:s:t]$, @TT "randomCurveP1P2"@
attempts to find a curve in $\PP^3$, which does not intersect the base locus of these projections.
If the curve did intersect the base locus the resulting curve in $\PP^1\times\PP^2$ would not have degree {\tt (d,d)}.
The number of attempts used to try to find such curves is controlled by the @TO [randomCurveP1P2, Attempt]@ option, which by default is set to 1000.
Example
randomCurveP1P2(3,0);
randomCurveP1P2(3,0,QQ);
Caveat
This creates a ring $F[x_{0,0},x_{0,1},x_{1,0},x_{1,1},x_{1,2}]$ in which the resulting ideal is defined.
///
doc ///
Key
Attempt
[randomCurveP1P2, Attempt]
Headline
limit number of attempts for randomCurveP1P2
Description
Text
When @TO "randomCurveP1P2"@ generates a random curve in $\PP^3$ using the @TO SpaceCurves@ package, it is possible the resulting
curve will intersect the base loci of the projections used to construct the curve in $\PP^1\times\PP^2$. If the curve
does intersect the base locusi it will generate a new random curve in $\PP^3$. The option @TT "Attempt"@ limits the number
of attempts to find a curve disjoint from the base loci before quitting. By default, Attempt is set to 1000.
SeeAlso
randomCurveP1P2
///
doc ///
Key
resolveViaFatPoint
(resolveViaFatPoint, Ideal, Ideal, List)
Headline
returns a virtual resolution of a zero-dimensional scheme
Usage
resolveViaFatPoint(I, irr, A)
Inputs
J:Ideal
saturated ideal corresponding to a zero-dimensional scheme
irr:Ideal
the irrelevant ideal
A:List
power you want to take the irrelevant ideal to
Outputs
:ChainComplex
virtual resolution of our ideal
Description
Text
Given a saturated ideal J of a zero-dimensional subscheme, irrelevant ideal irr, and a tuple A,
resolveViaFatPoint computes a free resolution of J intersected with A-th power of the irrelevant ideal.
See Theorem 4.1 of [BES20, @arXiv "1703.07631"@].
Below we follow example 4.7 of [BES20,@arXiv "1703.07631"@] and
compute the virtual resolution of 6 points in $\PP^1\times\PP^1\times\PP^2$.
Example
N = {1,1,2}
pts = 6
(S, E) = productOfProjectiveSpaces N
irr = intersect for n to #N-1 list (
ideal select(gens S, i -> (degree i)#n == 1)
);
I = saturate intersect for i to pts - 1 list (
P := sum for n to N#0 - 1 list ideal random({1,0,0}, S);
Q := sum for n to N#1 - 1 list ideal random({0,1,0}, S);
R := sum for n to N#2 - 1 list ideal random({0,0,1}, S);
P + Q + R
);
C = resolveViaFatPoint (I, irr, {2,1,0})
isVirtual(irr, C)
///
doc ///
Key
virtualOfPair
(virtualOfPair, Ideal, List)
(virtualOfPair, Module, List)
(virtualOfPair, ChainComplex, List)
Headline
creates a virtual resolution from a free resolution by keeping only summands of specified degrees
Usage
virtualOfPair(I, L)
virtualOfPair(M, L)
virtualOfPair(C, L)
Inputs
I:Ideal
ideal over multigraded ring
M:Module
module over multigraded ring
C:ChainComplex
free resolution of a module
L:List
multidegrees of summands to keep
Outputs
:ChainComplex
Description
Text
Given an ideal I or module M and a list of multidegrees L, this function produces a chain complex by iteratively
computing syzygies in degrees in L. In particular, if the list L contains only one element which is in the
multigraded regularity of M plus the dimension vector, the output will be the virtual resolution of a pair as
defined in Section 1 of [BES20]. See Algorithm 3.4 of [BES20, @arXiv "1703.07631"@] for further details.
If a resolution for the object exists in the cache or when the input is a chain complex C, virtualOfPair uses
this information by simply removing the summands in degrees not in L. This option is useful when a minimal free
resolution of M can be more efficiently computed in the engine or is already known. Otherwise, induced Schreyer
orders are used to speed up the computation of syzygies. Note that this speedup is often very significant.
When L contains more than one multidegree, summands with degrees in at least one member of L are kept.
For example, consider the ideal of three points in $\PP^1\times\PP^1$.
Example
X = toricProjectiveSpace(1) ** toricProjectiveSpace(1);
S = ring X; B = ideal X;
J = saturate(intersect(
ideal(x_1 - 1*x_0, x_3 - 4*x_2),
ideal(x_1 - 2*x_0, x_3 - 5*x_2),
ideal(x_1 - 3*x_0, x_3 - 6*x_2)),
B)
Text
We can now compute its minimal free resolution and a virtual resolution. One can show that $(2,0)$ is in the multigraded
regularity of this example. Thus, since we want to compute a virtual resolution we apply virtualOfPair to the element
$(3,1)$ since $(3,1)=(2,0)+(1,1)$ and $(1,1)$ is the dimension vector for $\PP^1\times\PP^1$.
Example
minres = res J
vres = virtualOfPair(J, {{3,1}}) --(3,1) = (2,0) + (1,1)
Text
Notice that the virtual resolution of the pair $(S^1/J, (2,0))$ is shorter and thinner than the graded minimal free
resolution of $S^1/J$.
Finally, we check that the result is indeed virtual.
Example
isVirtual(B, vres)
Caveat
Given an element of the multigraded regularity, one must add the dimension vector of the product of projective spaces
for this to return a virtual resolution.
///
doc ///
Key
[virtualOfPair, LengthLimit]
Headline
stop when the virtual resolution reaches this length
Description
Text
When the optional argument @TT "LengthLimit"@ is specified virtualOfPair will stop computing syzygies after the given
length is reached, otherwise computation continues until the resolution terminates.
SeeAlso
virtualOfPair
///
doc ///
Key
multigradedRegularity
(multigradedRegularity, Ring, Ideal)
(multigradedRegularity, Ring, Module)
(multigradedRegularity, NormalToricVariety, Ideal)
(multigradedRegularity, NormalToricVariety, Module)
[multigradedRegularity, Strategy]
[multigradedRegularity, LowerLimit]
[multigradedRegularity, UpperLimit]
LowerLimit
UpperLimit
Headline
computes the minimal elements of the multigraded regularity of a module over a multigraded ring
Usage
multigradedRegularity(S,I)
multigradedRegularity(S,M)
multigradedRegularity(X,I)
multigradedRegularity(X,M)
Inputs
S:Ring
a multigraded Cox ring
X:NormalToricVariety
a product of normal toric varieties
I:Ideal
an ideal over a multigraded ring
M:Module
a module over a multigraded ring
UpperLimit=>List
largest twist to compute cohomology for
LowerLimit=>List
smallest twist to compute cohomology for
Strategy=>String
implemented strategies are @TT "\"CohomologySearch\""@ and @TT "\"TruncationSearch\""@ (default)
Outputs
:List
a list of multidegrees
Description
Text
Given a module M over a multigraded ring S or a product of toric varieties X, this method finds the
minimal elements of the multigraded Castelnuovo-Mumford regularity of M as defined in Definition 1.1
of [MS04] (see @arXiv "math/0305214"@). If the input is an ideal, multigraded regularity of $S^1/I$ is computed.
There are two strategies implemented and run using @TO hooks@:
Tree
:@TT "Strategy => \"CohomologySearch\""@
:This strategy calls the @TO cohomologyHashTable@ method from @TO TateOnProducts@ and checks for the multidegrees where the Hilbert polynomial and Hilbert function match and where the higher sheaf cohomology vanishes.
:@TT "Strategy => \"TruncationSearch\""@ (default)
:This strategy uses @TO isQuasiLinear@ method from @TO LinearTruncations@ to search for the multidegrees where the module is regular by checking the Betti numbers of the truncation of the module. See Theorem 4.6 of @arXiv "2110.10705"@. This strategy is much faster.
Text
Note that both strategies require the module or ideal to be saturated by the irrelevant ideal of the Cox ring.
As an example, here we compute the minimal elements of the multigraded regularity for Example 1.4
of [BES20] (see @arXiv "1703.07631"@). We consider the example of a hyperelliptic curve of genus 4 in $\PP^1\times\PP^2$.
Example
X = toricProjectiveSpace(1)**toricProjectiveSpace(2)
S = ring X; B = ideal X;
I = ideal(x_0^2*x_2^2+x_1^2*x_3^2+x_0*x_1*x_4^2, x_0^3*x_4+x_1^3*(x_2+x_3))
Text
After saturating the defining ideal by the irrelevant ideal we may compute its multigraded regularity.
Example
J = saturate(I,B);
--debugLevel = 1
L = multigradedRegularity(X, J)
Text
If @TO "debugLevel"@ is larger than zero, additional information about the degree search is printed.
This method also accepts the ring provided by @TO productOfProjectiveSpaces@ from the @TO TateOnProducts@ package.
Contributors
Lauren Cranton Heller contributed to the code for this method.
Caveat
The input is assumed to be saturated.
Moreover, if the input is a module generated in non-positive degrees, then the output may be incorrect.
In that case, adding the optional argument
@PRE "LowerLimit => apply(n, i -> min(degrees M / (deg -> deg_i))) - dim X"@
where {\tt M} is the module and {\tt X} is the toric variety, may be a sufficient solution.
///