K3 polytopes and their quartic surfaces
jww G. Balletti and B. Sturmfels

Marta Panizzut

Summer Workshop on Lattice Polytopes
Osaka - August 01, 2018

Tropical hypersurfaces

K3 polytopes

Stablity of quartic surfacesTropical arithmetic

We work over the tropical semiring (R ∪ {∞}, ⊕, ).
Tropical operations are defined as follows:a ⊕ b = min{a, b} and ab = a + b.

They are associative and commutative.
Identity elements: a ⊕ ∞ = a and a

0 = a.

Multiplication is distributive with respect to addition.

Tropical polynomials
Let x1 , x2 , . . . , xn be variables representing elements in the tropicalsemiring. A monomial is any product of variables:
x1i1

x2i2···

xnin .

A polynomial is a finite linear combinations of monomials:
f (x1 , x2 , . . . , xn ) = ai

x1i1 x2i2 · · · xnin ⊕ aj

x1j1 x2j2 · · · xnjn ⊕ · · · .

We allow negative exponents.

RemarkPolynomials “translate” to piecewise-linear concave functions:f (x) = min{ai +i1 x1 +i2 x2 +· · ·+in xn , aj +j1 x1 +j2 x2 · · · jn xn , · · · }.

Tropical hypersurfaces
Given a polynomial f , we define the hypersurface T (f ) of f as theset of points x ∈ Rn at which the minimum is attained at least
twice.
Example:f = 3x 3 ⊕ 1x 2 y ⊕ 1xy 2 ⊕ 3y 3 ⊕ 1x 2 ⊕ xy ⊕ 1y 2 ⊕ 1x ⊕ 1y ⊕ 3

Tropical hypersurfaces

Theorem (Structure theorem)
The tropical hypersurface T (f ) is the support of a pure rational
polyhedral complex of dimension n − 1.
The closure of the connected components of the complement of atropical hypersurface T (f ) are called regions of T (f ). They areconvex polyhedra.

Tropical and classical hypersurfaces
Let K be an algebraically closed field with nontrivial valuation (forexample K = C{{t}}). Let f be an Laurent polynomial
Xf =cu x1u1 · · · xnun , with cu ∈ K .
u=(u1 ,...,un )∈Zn

We define its tropicalization trop(f ) ason
Xtrop(f ) = minn val(cu ) +ui xi .
u∈Zi≤nTheorem (Kapranov’s theorem)
The following sets coincide:1. {w ∈ Rn the min in trop(f )(w ) is attained at least twice};
2. the closure of {(val(y1 ), . . . , val(yn )) | (y1 , . . . , yn ) ∈ V (f )}.

Newton polytopes
L
Given a tropical polynomial f = v ∈Zn av x v , we define the Newton
polytope Newt(f ) as the polytope
Newt(f ) = conv(v : av 6= ∞).
Example: f = 3x 3 ⊕1x 2 y ⊕1xy 2 ⊕3y 3 ⊕1x 2 ⊕xy ⊕1y 2 ⊕1x ⊕1y ⊕3Newton polytopes
We consider the convex hull in Rn+1 of the points (v , av ). The
projection of the lower faces on Newt(f ) induces a subdivision of
the Newton polytope.
Example: f = 3x 3 ⊕1x 2 y ⊕1xy 2 ⊕3y 3 ⊕1x 2 ⊕xy ⊕1y 2 ⊕1x ⊕1y ⊕3Newton polytopes
We consider the convex hull in Rn+1 of the points (v , av ). The
projection of the lower faces on Newt(f ) induces a subdivision of
the Newton polytope.
Example: f = 3x 3 ⊕1x 2 y ⊕1xy 2 ⊕3y 3 ⊕1x 2 ⊕xy ⊕1y 2 ⊕1x ⊕1y ⊕3Newton polytopes and tropical hypersurfaces
Tropical hypersurfaces are dual to the regular subdivision of their
Newton polytopes induced by the coefficients.
Example: f = 3x 3 ⊕1x 2 y ⊕1xy 2 ⊕3y 3 ⊕1x 2 ⊕xy ⊕1y 2 ⊕1x ⊕1y ⊕3Smooth tropical hypersurfaces
A tropical hypersurface is smooth if the regular subdivision induced
by its coefficients is a unimodular triangulation, i.e., cells in the1.
subdivision are simplices of minimal volume n!
Examples:Unimodular

Not unimodular

Regions of a tropical quartic surface
Let 4∆3 be the 4-th dilatation of the standard simplex,4∆3 = conv (0, 0, 0), (4, 0, 0), (0, 4, 0), (0, 0, 4)
The point p = (1, 1, 1) is the unique interior lattice point of 4∆3 .

Let T (f ) be a smooth tropical hypersurface of degree 4 in R3 . Its
Newton polytope Newt(f ) is contained in 4∆3 . If Newt(f ) contains
p in its relative interior, the hypersurface cuts one bounded region
out.

Definition
A 3-dimensional polytope P is a K3 polytope if it arises as theclosure of the bounded region in the complement of a smooth
tropical surface of degree 4.

ExampleConsider the tropical quartic surface defined by the polynomial:

f = 5(x 4 ⊕ y 4 ⊕ z 4 ) ⊕ 3(x 3 y ⊕ x 3 z ⊕ xy 3 ⊕ y 3 z ⊕ xz 3 ⊕ yz 3 )⊕ 2(x 2 y 2 ⊕ x 2 z 2 ⊕ y 2 z 2 ) ⊕ 0(x 2 yz ⊕ xy 2 z ⊕ xyz 2 ) ⊕ 3(x 3 ⊕ y 3 ⊕ z 3 )⊕ 0(x 2 y ⊕ x 2 z ⊕ xy 2 ⊕ y 2 z ⊕ xz 2 ⊕ yz 2 ) ⊕ 2(x 2 ⊕ y 2 ⊕ z 2 )⊕ 0(xy ⊕ xz ⊕ yz) ⊕ 3(x ⊕ y ⊕ z) ⊕ (−9xyz) ⊕ 5.
The Newton polytope Newt(f ) is 4∆3 .

Let’s look at the K3 polytope defined by T (f )This is the smooth tropical quartic surface T (f ):Let’s look at the K3 polytope defined by T (f )This is the K3 polytope:

Its f -vector is (64, 96, 34).

The hunt for K3 polytopes

Smooth tropical quartic surfaces are dual to regular unimodular triangulations of their Newton polytopes. We switch our attention tothese objects.
From now on we will only talk about Newton polytopes P containedin 4∆3 containing p = (1, 1, 1) in their relative interior. They arecanonical polytopes.

The hunt for K3 polytopes

Warning: computing the secondary fan of 4∆3 does not seem feasible.
I 2∆3 has 10 lattice points and 15 regular triangulations,
I 3∆3 has 20 lattice points and 21 125 102 regulartriangulations.
Triangulations of 3∆3 were computed by Jordan, Joswig and Kastnerwith MPTOPCOM.

Central triangulations
We define the central part of a triangulation T as the subset of T
given by the simplices of T containing p. If T coincides with its
central part, then we say that T is central.

The K3 polytope is uniquely determined by the central part of thetriangulation T .

Central triangulations
We define the central part of a triangulation T as the subset of T
given by the simplices of T containing p. If T coincides with its
central part, then we say that T is central.

The K3 polytope is uniquely determined by the central part of thetriangulation T .
It is enough to consider central triangulations!

Central traingulations of canonical polytopes
How can we construct central triangulations of a canonical polytope?

{T central triangulation of P} ↔ {T triangulation of ∂P}

Central traingulations of canonical polytopes
How can we construct central triangulations of a canonical polytope?

{T central triangulation of P} ↔ {T triangulation of ∂P}
We are interested in polytopes P which admit at least one unimodular central triangulation.

Central traingulations of canonical polytopes
How can we construct central triangulations of a canonical polytope?

{T central triangulation of P} ↔ {T triangulation of ∂P}
We are interested in polytopes P which admit at least one unimodular central triangulation.
A polytope P is reflexive if Ap − c = 1, where Ax ≥ c are theequations defining P.
If P is reflexive,{T central unimod triang of P} ↔ {T is a unimod triang of ∂P}

Central traingulations of canonical polytopes
How can we construct central triangulations of a canonical polytope?

{T central triangulation of P} ↔ {T triangulation of ∂P}
We are interested in polytopes P which admit at least one unimodular central triangulation.
A polytope P is reflexive if Ap − c = 1, where Ax ≥ c are theequations defining P.
If P is reflexive,{T central unimod triang of P} ↔ {T is a unimod triang of ∂P}
A three dimensional canonical lattice polytope P is reflexive if andonly if every central fine triangulation of P is unimodular. We needto consider reflexive polytopes!

Theorem (Balletti-P-Sturmfels)

number of facets

Up to symmetry there are 15139 possible reflexive polytopes
contained in 4∆3 .
595

14300

945

1015
2025
number of lattice points

3035

1

Theorem (Balletti-P-Sturmfels)

number of facets

The reflexive polytopes of volume ≤ 30 in the theorem above admit
a total of 36 297 333 regular unimodular central triangulations.
Every K3 polytope with ≤ 30 vertices arises from one of these.
106

14

103

945

1015
20
25
number of lattice points

30

35

1

f -vectors of K3 polytopes
Theorem (Balletti-P-Sturmfels)
Let P be a K3 polytope obtained from a polytope P. Then P is asimple polytope. The entries of its f -vector arev = vol(P),f -vect(4, 6, 4)(6, 9, 5)(8, 12, 6)(10, 15, 7)(12, 18, 8)(14, 21, 9)(16, 24, 10)(18, 27, 11)(20, 30, 12)# of P
9102
412
959
1642
2083
2194
1997
1646

e=3vol(P),2f -vect(22, 33, 13)(24, 36, 14)(26, 39, 15)(28, 42, 16)(30, 45, 17)(32, 48, 18)(34, 51, 19)(36, 54, 20)(38, 57, 21)

f = |P ∩ Z2 | − 1.

# of P
1248
922
628
465
295
203
128
85
53

f -vect(40, 60, 22)(42, 63, 23)(44, 66, 24)(46, 69, 25)(48, 72, 26)(50, 75, 27)(54, 81, 29)(56, 84, 30)(64, 96, 34)# of P
27
18
7
92
2
1
11

Moduli space of quartic surfaces
A quartic surface is the variety in P3 defined by a homogeneouspolynomial of degree 4 in C[x, y , z, w ],X
f (x, y , z, w ) =cijk x i y j z k w 4−i−j−k .
i+j+k≤4

Moduli space of quartic surfaces
A quartic surface is the variety in P3 defined by a homogeneouspolynomial of degree 4 in C[x, y , z, w ],X
f (x, y , z, w ) =cijk x i y j z k w 4−i−j−k .
i+j+k≤4

The 35 coefficients cijk parameterize all the quartic surfaces. So
we consider the 34-dimesional projective space HS4,3 = P(HS4,3 ) ofquartic surfaces. This gives us a “moduli space of quartic surfaces”.

Moduli space of quartic surfaces
A quartic surface is the variety in P3 defined by a homogeneouspolynomial of degree 4 in C[x, y , z, w ],X
f (x, y , z, w ) =cijk x i y j z k w 4−i−j−k .
i+j+k≤4

The 35 coefficients cijk parameterize all the quartic surfaces. So
we consider the 34-dimesional projective space HS4,3 = P(HS4,3 ) ofquartic surfaces. This gives us a “moduli space of quartic surfaces”.
More precisely, the special linear group SL(4) acts on HS4,3 , andon the associated polynomial ring C[HS4,3 ], generated by cijk . Themoduli space of quartic surfaces in P3 is the projective variety determined by Proj C[HS4,3 ]SL(4) .

Stable elements

In the context of Geometric Invariant Theory, we want to restricto a “nicer” set inside the moduli space. This is the set of stableelements.

Stable elements

In the context of Geometric Invariant Theory, we want to restricto a “nicer” set inside the moduli space. This is the set of stableelements.
More precisely, f is stable if the orbit O(f )SL(4) is closed and thestabilizer stab(f ) is finite.

Stable elements

In the context of Geometric Invariant Theory, we want to restricto a “nicer” set inside the moduli space. This is the set of stableelements.
More precisely, f is stable if the orbit O(f )SL(4) is closed and thestabilizer stab(f ) is finite.
Determining whether an element is stable is connected to the studyof its singular locus.

Theorem (Shah ’81)
If the singular locus of a quartic surface contains at most rational
double points, then the surface is stable.

Arnold’s classification ’72
Ak : x 2 + y 2 + z k+1

Dk : x 2 + y 2 z + z k−1

Pictures from Greuel-Lossen-Shustin “Introduction to Singularities
and Deformation”.

Arnold’s classification ’72

E6 : x 2 + y 3 + z 4 , E7 : x 2 + y 3 + yz 3 , and E8 : x 2 + y 3 + z 5

Pictures from Greuel-Lossen-Shustin “Introduction to Singularities
and Deformation”.

A combinatorial criterion
Theorem (Mumford ’77)
A point f in HS4,3 is stable if and only if, for every choice of
coordinates, and for all planes H through p, each open halfspace of
H contains a monomial of f .

A combinatorial criterion
Theorem (Mumford ’77)
A point f in HS4,3 is stable if and only if, for every choice of
coordinates, and for all planes H through p, each open halfspace of
H contains a monomial of f .

A reflexive lattice polytope P contained in 4∆3 is called minimal if
it does not properly contain any reflexive polytope.
There are precisely 115 minimal reflexive polytopes in 4∆3 .

Stability

Theorem (Balletti-P-Sturmfels)
Let f ∈ C[x, y , z, w ] be a generic homogeneous quartic surface
whose Newton polytope arises from a smooth tropical surface.
Then the quartic surface V (f ) in P3 is stable.

Stability

Theorem (Balletti-P-Sturmfels)
Let f ∈ C[x, y , z, w ] be a generic homogeneous quartic surface
whose Newton polytope arises from a smooth tropical surface.
Then the quartic surface V (f ) in P3 is stable.
I We show the stability of surfaces having a minimal polytope
as Newton polytope by studying their singular locus.

Stability

Theorem (Balletti-P-Sturmfels)
Let f ∈ C[x, y , z, w ] be a generic homogeneous quartic surface
whose Newton polytope arises from a smooth tropical surface.
Then the quartic surface V (f ) in P3 is stable.
I We show the stability of surfaces having a minimal polytope
as Newton polytope by studying their singular locus.
I We use Mumford’s criterion to conclude that also genericsurfaces with Newton polytope containing a minimal one arestable.

Thank you!