The GIT Compactification of Quintic Threefolds

The GIT Compactification of Quintic Threefolds

Chirag Lakhani Department of Mathematics & Computer Science
High Point University
High Point, NC, 27262
USA
clakhani@highpoint.edu
Abstract.

In this article, we study the geometric invariant theory (GIT) compactification of quintic threefolds. We study singularities, which arise in non-stable quintic threefolds, thus giving a partial description of the stable locus. We also give an explicit description of the boundary components and stratification of the GIT compactification.

1. Introduction

Quintic threefolds are some of the simplest examples of Calabi-Yau varieties. Physicists have given Calabi-Yau varieties a great deal of attention, in the last 30 years, because they give the right geometric conditions for some superstring compactifications [CHSW85]. In mirror symmetry, in particular, the Kahler moduli space and complex structure moduli space of Calabi-Yau varieties are important objects of study. The purpose of this paper is to describe the complex structure moduli space of quintic threefolds using geometric invariant theory (GIT).

GIT constructions of moduli spaces of projective varieties are automatically projective, therefore have a natural compactification. The Hilbert-Mumford criterion provides a numerical tool, which is useful when constructing moduli spaces using GIT. Despite having this tool, it is still difficult to construct moduli spaces in dimension 2 or higher. There are a some cases where such moduli spaces have been constructed, such as degree 2 and degree 4 surfaces by Shah [Sha80, Sha81], cubic threefolds by Allcock and Yokoyama [All03, Yok02], and cubic fourfolds by Laza [Laz09].

A quintic threefold is the zero set of a homogeneous degree form . The space represents the set of degree forms in . The parameter space of quintic threefolds is then represented by , which is the projectivization of the space of coefficients of quintic forms . and both have natural -actions. Two threefolds are equivalent if one form can be transformed into another by an -action. In order to construct the moduli space using GIT, the stable and semistable quintic threefolds must be identified. A quintic threefold is semistable if there is a -invariant function on where does not vanish. A semistable quintic threefold is stable if its -orbit, in the space of semistable quintic threefolds, is closed and the isotropy group of is finite. The space of semistable quintic threefolds is denoted and the space of stable quintic threefolds is denoted . The -orbits of threefolds in are closed, so the quotient forms an orbit space. The addition of semistable quintic threefolds compactifies the moduli space by making it a projective variety i.e. is projective. Two semistable quintic threefolds map to the same point in if their closures satisfy

(1)

This establishes an orbit-closure relationship for semistable quintic threefolds where if they satisfy the property 1. Furthermore, all threefolds in the same orbit-closure equivalence class map to the same point in . Every orbit-closure equivalence class in has a unique closed orbit representative called the minimal orbit. The boundary components of are represented precisely by these minimal orbits.

Remark 1.

We will follow the terminology in GIT [MFK94]. Unstable will mean not semistable, non-stable will mean not stable, and strictly semistable will mean semistable but not stable.

The main results of this paper are a description of the non-stable quintic threefolds in terms of singularities, partial description of the stable locus, and a complete description of boundary components and stratification of the GIT compactification using minimal orbits. The first main result of the paper is given in section 2.3 which describes the non-stable quintic threefolds in terms of singularities.

Theorem 2.

A quintic threefold is non-stable if and only if one of the following properties holds:

  1. contains a double plane;

  2. is a reducible variety, where a hyperplane is one of the components;

  3. contains a triple line;

  4. contains a quadruple point;

  5. contains a triple point with the following properties:

    1. the tangent cone of is the union of a double plane and another hyperplane;

    2. the line connecting a point in the double plane with the triple point has intersection multiplicity 5 with ;

  6. has a double line where every point has the following properties:

    1. the tangent cone of each point is a double plane ;

    2. each point has the same double plane tangent cone i.e. for some double plane ;

    3. the line connecting the point on the tangent cone and a point has intersection multiplicity 4 with ;

  7. contains a triple point and a plane with the following properties:

    1. the tangent cone of contains a triple plane of ;

    2. the singular locus of , when restricted to , is the intersection of two quartic curves and ;

    3. the point is a quadruple point of and .

A partial description of the stable locus is given in section 2.4. The analysis of singularities of the non-stable quintic threefolds gives a partial description of the singularities that occur in the stable locus. In particular, all smooth quintic threefolds and quintic threefold with at worst singularities will be GIT stable. Following the approach of Laza [Laz09], the minimal orbits can be explicitly described using Luna’s criterion[Lun75, VP89]. In order to find the minimal orbits, the non-stable quintic threefolds degenerate into families of quintic threefolds given by equations 4-7, which are denoted the first level of minimal orbits. Luna’s criterion determines which members of these families represent closed orbits, this is done in section 3.2. Certain hypersurfaces in these families are unstable and therefore represent unstable quintic threefolds or they degenerate, even further, into a family of quintic threefolds given by equations 8-17. The families represented by equations 8-17 are called the second level of minimal orbits. The second level of minimal orbits represent how the boundary strata of the components in equations 4-7 intersect. Applying Luna’s criterion to the second level of minimal orbits will determine which quintic threefolds are closed orbits, unstable orbits, and which hypersurfaces degenerate even further. It will be shown in section 4.1 that non-closed orbits in the second level of minimal orbits will eventually degenerate to the hypersurface , which is the hypersurface with normal crossings singularities. This completely determines the boundary structure and stratification of the GIT compactification of the moduli space of quintic threefolds.

Section 2 is devoted to the combinatorics and geometrical study of non-stable quintic threefolds. Using the Hilbert-Mumford criterion, a combinatorial study of the monomials to be included in maximal non-stable families of quintic threefolds will be done. Each maximal non-stable family has a destabilizing one-parameter subgroup (1-PS) which gives rise to a “bad flag” that picks out the worst singularities in the family. This is used to prove Theorem 2. The last part of section 2 gives a partial description of the stable locus. Section 3 introduces Luna’s criterion and then applies it to study the closed orbits in equations 4-7. Section 4 studies the closed orbits and further degenerations of equations 8-17, thereby giving a complete description of the boundary stratification of the GIT compactification of quintic threefolds.

Acknowledgements

The author would like to Amassa Fauntleroy and Radu Laza for their advice and support in writing this paper. The author would also like to thank Ryan Therkelsen for introducing the author to Stembridge’s poset Maple package.

2. Maximal Nonstable Families

The Hilbert-Mumford criterion is an important tool when establishing stability and semistability in GIT.

Remark 3.

Following the convention in [Laz09], a normalized 1-PS is a map , where is the the standard maximal torus of , with the additional property that satisfy and .

For a quintic form and a normalized 1-PS , the numerical function is defined as follows,

(2)

The Hilbert-Mumford criterion states a quintic form f is stable (semistable) if and only if for every 1-PS the numerical function . It can be restated so that a quintic form is non-stable (unstable) if there exists a 1-PS where . If quintic forms are analyzed up to coordinate transformation (-action), then the -equivariance of the numerical function[LR08],

(3)

restricts to check checking the criterion for only normalized 1-PS in the standard maximal torus of .

Remark 4.

In the remainder of the paper the a monomial will also be written as a vector denoted . Also, the following combinatorial procedure described below is based on similar combinatorial techniques described in Mukai [Muk03] and applied by Laza [Laz09] in the case of cubic fourfolds.

The normalized 1-PS induce a partial order on the set of quintic monomials given by

The following lemma is useful in creating an algorithm to determine the poset structure of quintic monomials.

Lemma 5 (c.f. [Muk03] p.225).

For two monomials and

This criterion is useful because one can directly check whether two monomials are related in the poset by checking the subsequent inequalities. Using Maple[MGH05], the above criterion can be used to find all partial order relationships between monomials. Stembridge’s poset package for Maple [Ste98] is used to find the minimal covering relationships for these monomials and thereby creating the poset for quintic monomials. The code for this entire procedure is given in appendix A. The figure for this poset structure is given in figure 3 at the end of the paper.

2.1. Combinatorics of Non-Stable Families

The poset structure on quintic monomials greatly simplifies the Hilbert-Mumford criterion analysis on quintic polynomials . For a fixed normalized 1-PS and a monomial , does not change when any monomials below it in the poset are added to . For a fixed normalized 1-PS , () represent the set of monomials in the poset where (). By the Hilbert-Mumford criterion, the monomials of every non-stable quintic polynomial , up to coordinate transformation, belong to a family of the form . The maximal non-stable families, denoted , are the largest possible families of the form . The corresponding of is called the family’s destabilizing 1-PS. From the poset structure, there will be a finite number of maximal non-stable families . Any non-stable quintic form , up to coordinate transformation; will be long to one of these families.

The procedure for determining the set of maximal non-stable families would be to start from the top monomial and work down the poset until one finds a monomial which has a normalized 1-PS where . By restricting to normalized one can use linear programming to determine whether such a exists for a particular monomial. The linear programming script is given in appendix B. Using this procedure it is determined that the top most monomials which have are [3,0,0,2,0], [4,0,0,0,1], [2,0,3,0,0], and [1,4,0,0,0], the poset of monomials below these top monomials are denoted SS1, SS2, SS3, and SS4. The figures for these posets are given in figures 456, and 7 at the end of the paper. The destabilizing 1-PS is given in table 1. Other maximal non-stable families are found by finding the top-most monomials in the families which have a common destabalizing 1-PS . There are three other such families denoted . These three families have multiple maximal monomials. The figures for these posets are given in figures 89101112 and 13 at the end of the paper.

The combinatorial procedure above determines the maximal families where . By the Hilbert-Mumford criterion, every semistable or unstable hypersurface will transform, via a coordinate transformation, into one of maximal non-stable families .

Remark 6.

represents polynomials which are a linear combination of degree monomials in and multiplied by a degree monomials in and

Proposition 7.

is non-stable if and only if it belongs, via a coordinate transformation, to a hypersurface in families found in table 1.

Family Destabilizing 1-PS Subgroup Maximal Monomial Degeneration
SS1
SS2
SS3
SS4
SS5
SS6
SS7
Table 1. Nonstable Families SS1 - SS7

For any non-stable , the -orbit will not necessarily be closed. Using the destabilizing 1-PS , the closure = is a quintic form invariant with respect to . The forms and will map to the same point in the GIT quotient. If the orbit of is closed then it is a minimal orbit.

(4)
(5)
(6)
(7)

In Section 3, it will be shown that a generic member of one of the families will represent a minimal orbit. Certain hypersurfaces in will degenerate further into a member of one of the families .

(8)
(9)
(10)
(11)
(12)
(13)
(14)
(15)
(16)
(17)

2.2. Bad Flags

The maximal non-stable families will be characterized in terms of singularities found on a generic member of one of these families. A destabilizing 1-PS has an associated “bad flag” of the vector spaces . A general principal, given by Mumford [MFK94], states that these “bad flags” pick out the singularities which cause the family to become semistable or unstable.

Using the approach given by Laza [Laz09] it can be shown that a 1-PS gives a weight decomposition of based on the eigenvalues of acting on .

Definition 8.

For a 1-PS let be a subset of which have the same weights and let be the weight.

(18)

The standard flag is

(19)
Definition 9.

Given a 1-PS let , , represent the collection of common weights of . Let be ordered by increasing value of weights (i.e. has lowest weight). The associated flag for is

(20)

This is a subflag of the standard flag (20).

For the maximal destabilizing families the associated “bad flags” are

Family Destabilizing 1-PS Subgroup Destabilizing Flag
SS1

SS2

SS3

SS4

SS5

SS6

SS7



Table 2. Destabilizing Flags of SS1-SS7

2.3. Geometric Interpretation of Maximal Semistable Families

In order to determine the singularities which occur on threefolds in families , we intersect the general form of the equation with its associated destabilizing flag. This will give some description of the types of singularities, which occur in each family. A precise description of each such family is given in the propositions below. Some of the singularity analysis is based on describing the tangent cone and intersection multiplicities of the tangent cone at singular points, a detailed introduction of these topics is given in Beltrametti et al. ([BCGMB09] ch.5)

Proposition 10.

A hypersurface is of type SS1 if and only if contains a double plane.

Proof.

Let be of type then it is equivalent, via a coordinate transformation, to the hypersurface

(21)

This hypersurface contains the ideal which is a double plane in .

Let be a hypersurface which contains a double plane. By a coordinate transformation we can assume the double plane is . The most general equation which contains the ideal is (21).

Proposition 11.

A hypersurface is of type SS2 if and only if is a reducible variety, where a hyperplane is one of the components. In particular, the singularity is the intersection of the hyperplane with the other component, which is generically a degree 4 surface.

Proof.

Let be of type then it is equivalent, via a coordinate transformation, to the hypersurface

(22)

This hypersurface has the hyperplane as a component.

Let be a reducible hypersurface where a hyperplane is a component. The polynomial defining can be factored into , where is a degree 1 polynomial. By a coordinate transformation we can map the hyperplane defining to . Without loss of generality . Since since is of degree 5 then by neccesity is of degree 4 therefore is of the form (22).

Proposition 12.

A hypersurface is of type SS3 if and only if contains a triple line.

Proof.

Let be of type then it is equivalent, via a coordinate transformation, to the hypersurface

(23)

This hypersurface contains the ideal which is a triple line in .

Let be a hypersurface which contains a triple line. By a coordinate transformation, we can assume the triple line is . The most general equation which contains the ideal is (23).

Proposition 13.

A hypersurface is of type if and only if contains a quadruple point.

Proof.

Let be of type then it is equivalent, via a coordinate transformation, to the hypersurface

(24)

This hypersurface contains the ideal which is a quadruple point in .

Let be a hypersurface which contains a quadruple point. By a coordinate transformation we can assume the quadruple point is . The most general equation which contains the ideal is (24).

Proposition 14.

A hypersurface is of type if and only if has a triple point with the following properties:

  • the tangent cone of is the union of a double plane and another hyperplane;

  • the line connecting a point in the double plane with the triple point has intersection multiplicty 5 with the hypersurface.

Proof.

Let be of type then it is equivalent, via a coordinate transformation, to the hypersurface

(25)

This hypersurface contains the triple point . The tangent cone is the hypersurface defined by

(26)

which is the union of a double hyperplane and another general hyperplane . The points whose lines passing through the triple point which have intersection multiplicity 5 with the hypersurface, is the locus of and . Since is a component of both terms then a line emanating from the hyperplane to the triple point will have multiplicity 5.

Let be a hypersurface which contains a triple point. By a coordinate transformation we can assume the triple point is . The most general equation which contains the ideal is

(27)

If the tangent cone is the union of a double plane and another hyperplane then

(28)

where and are linear forms. By a coordinate transformation which keeps the triple point fixed we can map the hyperplane to . So without loss of generality

(29)

If a general line from the hyperplane to the triple point has multiplicity 5 then

(30)

This occurs only if has as a component so

(31)

which is precisely of the form (25).

Proposition 15.

A hypersurface is of type if and only if has a double line where every point has the following properties:

  • the tangent cone of each point is a double plane ;

  • each point has the same double plane tangent cone i.e. for some double plane ;

  • the line connecting the point on the tangent cone and a point has intersection multiplicity 4 with the hypersurface.

Proof.

Let be of type then it is equivalent, via a coordinate transformation, to the hypersurface

(32)

This hypersurface contains the double line . For any point of the double line the tangent cone is the same double plane given by . The points which have intersection multiplicity 4 with the double line are the locus of and . Since is a component of both terms then the line emanating from the hyperplane to any point of the double line will have multiplicity 4.

Let be a hypersurface which contains a double line. By a coordinate transformation we can assume the double line is . The most general equation which contains the ideal is

(33)

If the tangent cone at every point on the double line is the same double plane then

(34)

where is a linear form. By a coordinate transformation, which keeps the double line fixed, the hyperplane is mapped to . So without loss of generality,

(35)

If the line going from the hyperplane to any point of the double line has multiplicity 4 then

(36)

This occurs only if has as a component so

(37)

which is precisely of the form (32).

Proposition 16.

A hypersurface is of type if and only if contains a triple point and a plane , where has the following properties:

  • the tangent cone of contains a triple plane of ;

  • the singular locus of , when restricted to , is the intersection of two quartic curves and ;

  • the point is a quadruple point of and .

Proof.

Let be of type then it is equivalent, via a coordinate transformation, to the hypersurface

(38)

This hypersurface contains the triple point given by the ideal and a plane given by . The tangent cone is the hypersurface defined by which which contains the triple plane of . When the differential of is restricted to the plane the only non-trivial contribution comes from the term

(39)