A 1-dimensional component of K-moduli of del Pezzo surfaces
AA 1-DIMENSIONAL COMPONENTOF K-MODULI OF DEL PEZZO SURFACES
ANDREA PETRACCI
Abstract.
We explicitly construct a component of the K-moduli space ofK-polystable del Pezzo surfaces which is a smooth rational curve. Introduction
One of the most important and recent results in K-stability and in the theoryof Fano varieties is the construction of K-moduli [3, 8–10, 14, 17, 21, 33, 34]. It hasbeen proved that, for every positive integer n and every positive rational number V , Q -Gorenstein families of K-semistable Fano C -varieties of dimension n and anti-canonical volume V form an algebraic stack M Kss n,V of finite type over C ; moreover,this stack admits a good moduli space M Kps n,V , which is a projective scheme over C , and the set of C -points of M Kps n,V coincides with the set of K-polystable Fanovarieties of dimension n and anticanonical volume V . We refer the reader to [32]for a survey on these topics.The case of smoothable del Pezzo surfaces has been extensively studied [22, 24,25]. Moreover, K-moduli are understood for cubic 3-folds [20], cubic 4-folds [18],and for certain pairs (surface, curve) [5, 6].The goal of this note is to show how toric geometry and deformation theory canhelp understanding the geometry of explicit components of K-moduli. Similar ideaswere used in [15] to construct examples of reducible or non-reduced K-moduli ofFano 3-folds, in [19] to study the K-stability of certain del Pezzo surfaces with Fanoindex 2, and in [23] to study the dimension of K-moduli. In this note we analyse aspecific example of K-polystable toric del Pezzo surface and we prove the following: Theorem 1.1.
There exists a connected component of M Kps2 , which is isomorphicto P . It is natural to wonder about the following:
Question 1.2.
Does there exist V ∈ Q > such that a connected component of M Kps2 ,V is a smooth curve of positive genus? Outline. In § § P and a K-polystable toric del Pezzo surface X , and we analyse itsdeformation theory; in particular, we show that the connected component of theK-moduli space of K-polystable del Pezzo surfaces that contains X is smooth and1-dimensional. In § X is a hypersurface in a toric 3-fold Y and in § X inside the linear system |O Y ( X ) | on Y gives theversal deformation of X . This gives a non-constant morphism from an open subset a r X i v : . [ m a t h . AG ] F e b ANDREA PETRACCI of |O Y ( X ) | to the K-moduli space. In § § Notation and conventions.
We work over an algebraically closed field of charac-teristic zero, which is denoted by C . A Fano variety is a normal projective varietyover C such that its anticanonical divisor is Q -Cartier and ample. A del Pezzo surface is a Fano variety of dimension 2. Every toric variety we consider is normal.If r, a , . . . , a n are integers and r ≥
1, then the symbol r ( a , . . . , a n ) standsfor the quotient of A n under the action of the cyclic group µ r defined by ζ · ( x , . . . , x n ) = ( ζ a x , . . . , ζ a n x n ). We use the same symbol to indicate the ´etale-equivalence class of the singularity of this quotient variety at the image of the originof A n . Acknowledgements.
The author learnt most of the techniques and the ideasdescribed in this note during countless conversations with Tom Coates, AlessioCorti, Al Kasprzyk and Thomas Prince over the years; it is a pleasure to thankthem. 2.
Proof
Deformations of (1 , . The cyclic quotient singularity (1 ,
1) is the affinecone over the 4th Veronese embedding of P into P . The deformations of thissingularity have been studied by Pinkham [29]. Here we concentrate on the Q -Gorenstein deformation.The index 1 cover of (1 ,
1) is (1 , xy − z = 0) in A x,y,z . Since the miniversal deformation of (1 ,
1) is given by xy − z + t = 0 in A x,y,z over C [[ t ]], we have that the miniversal Q -Gorensteindeformation of (1 ,
1) is given by xy − z + t = 0 in (1 , , x,y,z over C [[ t ]]. Thisspecifies a formal morphism(1) Spf( C [[ t ]]) −→ Def qG (cid:18)
14 (1 , (cid:19) , which is smooth and induces an isomorphism on tangent spaces. We will always usethis morphism when considering the Q -Gorenstein deformation functor of (1 , § xy − z + s + s z = 0 in (1 , , x,y,z over C [[ s , s ]]. By versality we get formalmorphisms(2) Spf( C [[ s , s ]]) −→ Spf( C [[ t ]]) (1) −→ Def qG (cid:18)
14 (1 , (cid:19) . Via the automorphism of (1 , , x,y,z × Spf( C [[ s , s ]]) given by z (cid:55)→ z (cid:112) − s z , we get an isomorphism of the given deformation with xy − z + s = 0, which isexactly the miniversal deformation. Therefore the morphism in (2) is induced bythe local C -algebra homomorphism C [[ t ]] → C [[ s , s ]] given by t (cid:55)→ s . Figure 1.
The polygon P in § The surface X . In the lattice N = Z consider the polygon P which is theconvex hull of the points (cid:18) (cid:19) , (cid:18) (cid:19) , (cid:18) − (cid:19) , (cid:18) − − (cid:19) , (cid:18) − − (cid:19) , (cid:18) − (cid:19) and is depicted in Figure 1. (The meaning of the red segments in this figure will beclear in § X be the toric del Pezzo surface associated to the face fan (alsocalled spanning fan) of P .Let P ◦ denote the polar of P ; it is a rational polytope in the dual lattice M =Hom Z ( N, Z ) and is the moment polytope of the toric boundary of X , which is ananticanonical divisor. The anticanonical volume of X is the normalised volume of P ◦ , which is . Since P is centrally symmetric (i.e. P = − P ), also P ◦ is centrallysymmetric, hence the barycentre of P ◦ is the origin. Therefore X is K-polystableby [7].By analysing the cones in the face fan of P one can see that the singularitiesof X are as follows: 2 points of type (1 , (1 , (1 , (1 ,
1) and (1 ,
2) are Q -Gorenstein rigid, i.e. theydo not deform. The Q -Gorenstein deformations of (1 ,
1) have been considered in § Q -Gorenstein defor-mations of X , so the Q -Gorenstein smoothings of the two (1 ,
1) points of X , whichwe denote p and p , can be realised globally and simultaneously. More precisely,since H i ( X, T X ) = 0 for i ≥ p i ∈ X )(3) Def qG ( X ) −→ Def qG ( p ∈ X ) × Def qG ( p ∈ X )is smooth and induces an isomorphism on tangent spaces. So C [[ t , t ]] is the hullof Def qG ( X ) and t i is the Q -Gorenstein smoothing parameter of ( p i ∈ X ). In thenext section we will realise the miniversal Q -Gorenstein deformation of X in a linearsystem in a toric Fano 3-fold.Let T N denote the 2-dimensional torus N ⊗ Z G m = Spec C [ M ] which acts on X .The automorphism group of the polygon P has order 2 and is generated by − id N .Since every facet of P ◦ has no interior lattice points, by [15, Proposition 2.6] theautomorphism group of X is G = T N (cid:111) C , where C is the cyclic group of order 2. ANDREA PETRACCI
The automorphism group G acts on the hull C [[ t , t ]]. The weights of t (resp. t ) in M is (0 ,
1) (resp. (0 , − C [[ t , t ]] T N is C [[ t t ]]. The group C swaps t and t , so it leaves t t invariant. Therefore theinvariant subring C [[ t , t ]] G is C [[ t t ]].Let M (resp. M ) be the connected component of M Kss2 , (resp. M Kps2 , ) whichcontains the point corresponding to X . By the Luna ´etale slice theorem for algebraicstacks [4] the local structure of M → M is given by the cartesian square[Spec C [[ t , t ]] / G ] (cid:15) (cid:15) (cid:47) (cid:47) M (cid:15) (cid:15) Spec C [[ t t ]] (cid:47) (cid:47) M where the horizontal maps are formally ´etale and maps the closed point to [ X ].Since M is normal and projective, we have that M is a smooth projective curve.2.3. X is a divisor in a toric Fano -fold. Here we apply the Laurent inversionmethod [13, 30, 31] to construct a toric Fano 3-fold Y where X lies as a divisor.Let e , e be the standard basis of N = Z . Consider the decomposition N = N ⊕ N U where N = Z e and N U = Z e . Let M be the dual lattice of N . Let Z be the T M -toric variety associated to complete fan in the lattice M with rays generated by e ∗ and − e ∗ . It is clear that Z is isomorphic to P . Let Div T M ( Z ) be the rank-2 latticemade up of the torus invariant divisors on Z : a basis of Div T M ( Z ) is given by E + , E − , the torus invariant divisors on Z associated to the rays e ∗ , − e ∗ respectively.The divisor sequence of Z is0 −→ N = Z e ρ (cid:63) = − −−−−−−−→ Div T M ( Z ) = Z E + ⊕ Z E − (cid:16) (cid:17) −−−−−→ Pic( Z ) = Z −→ . We consider the following ample torus invariant divisors on ZD x = E + + E − D y = − E + + 2 E − D x = E + + E − D y = 2 E + − E − and their corresponding moment polytopes in N : P D x = conv {− e , e } P D y = conv { e , e } ,P D x = conv {− e , e } P D y = conv {− e , − e } . Now consider the following elements in the lattice N U = Z e : χ x = 2 e χ y = e χ x = − e χ y = − e . The polytopes P D x + χ x P D y + χ y P D x + χ x P D y + χ y in N = N ⊕ N U are the four red segments in Figure 1. Clearly the polygon P isthe convex hull of these four segments. By [13, Definition 3.1] the set S = { ( D x , χ x ) , ( D x , χ x ) , ( D y , χ y ) , ( D y , χ y ) } is a scaffolding on the Fano polygon P .Consider the rank-3 lattice ˜ N := Div T M ( Z ) ⊕ N U = Z E + ⊕ Z E − ⊕ Z e . Let ˜ M be the dual lattice of ˜ N and let (cid:104)· , ·(cid:105) : ˜ M × ˜ N → Z be the duality pairing. Following[13, Definition A.1] we consider the polytope Q S ⊆ ˜ M R defined by the followinginequalities: (cid:104) · , − D x + χ x (cid:105) ≥ − , (cid:104) · , − D x + χ x (cid:105) ≥ − , (cid:104) · , − D y + χ y (cid:105) ≥ − , (cid:104) · , − D y + χ y (cid:105) ≥ − , (cid:104) · , E + (cid:105) ≥ , (cid:104) · , E − (cid:105) ≥ . Let Σ S be the normal fan of Q S . One can see that Σ S is the complete simplicialfan in ˜ N = Div T M ( Z ) ⊕ N U with rays generated by the following vectors: x = − D x + χ x = − E + − E − + 2 e x = − D x + χ x = − E + − E − − e y = − D y + χ y = E + − E − + e y = − D y + χ y = − E + + E − − e z = E + ,z = E − . Let Y be the T ˜ N -toric variety associated to the fan Σ S . Thus Y is a Q -factorialFano 3-fold with Cox coordinates x , x , y , y , z , z . With respect to the basis of˜ N given by E + , E − , e , the ray map Z → ˜ N is given by the matrix − − − − − − − − . By computing the kernel of this matrix, one finds that the divisor map Z → Cl( Y ) (cid:39) Z is given by the following matrix. x x y y z z L L L This matrix gives the weights of an action of the torus G on A ; Y is the GITquotient of this action with respect to the stability condition given by the irrelevantideal ( x , x , z ) · ( x , x , z ) · ( y , y ) · ( y , z ) · ( y , z ) . Here L , L , L denote the elements of the chosen basis of Cl( Y ).We now consider the injective linear map θ := ρ (cid:63) ⊕ id N U : N = N ⊕ N U −→ ˜ N = Div T M ( Z ) ⊕ N U . ANDREA PETRACCI
By [13, Theorem 5.5] θ induces a toric morphism X → Y which is a closed embed-ding. We want to understand the ideal of this closed embedding in the Cox ring of Y by using the map θ .We follow [31, Remark 2.6]. We see that θ ( N ) is the hyperplane defined by thevanishing of h = E ∗ + + E ∗− ∈ ˜ M . Now we compute the duality pairing between h and the primitive generators of the rays of Σ S : (cid:104) h, x (cid:105) = (cid:104) h, x (cid:105) = − (cid:104) h, y (cid:105) = (cid:104) h, y (cid:105) = − (cid:104) h, z (cid:105) = (cid:104) h, z (cid:105) = 1. We get that the polynomial(4) z z − y y x x is the generator of the ideal of the closed embedding X (cid:44) → Y in the Cox ring of Y .In other words, X is the hypersurface in Y defined by this polynomial in the Coxcoordinates of Y .2.4. Deformations of X . We have seen that X is a particular hypersurface in Y in the linear system | L + 6 L + 6 L | . We see that H ( Y, L + 6 L + 6 L ) hasdimension 4 and its monomial basis is made up of the monomials z z , y y x x , x y , x y . We perturb the equation (4) and get(5) z z − y y x x + s x y + s x y . This defines a flat family X → A = Spec C [ s , s ] of hypersurfaces in the linearsystem | L + 6 L + 6 L | . The fibre over the origin is clearly X .We want to show that, after base change to C [[ s , s ]], this family is the miniversal Q -Gorenstein deformation of X . Since the map in (3) is smooth and induces anisomorphism on tangent spaces, we need to check that locally this family inducesthe miniversal deformations of the singularity germs of X . Let t and t be thetwo smoothing parameters of the two (1 ,
1) singularities of X . We proceed byanalysing each chart of the affine open cover of Y given by the fan Σ S . • The cone σ x ,z ,z gives the isolated singularity (1 , , x ,z ,z on Y . Inthis chart, by dehomogenising (5), we get the equation z z − x + s x + s =0 in the orbifold coordinates. This is exactly the Q -Gorenstein smoothingof (1 ,
1) described in § t = s . • The cone σ x ,z ,z gives the isolated singularity (1 , , x ,z ,z on Y . Inthis chart we get the equation z z − x + s + s x = 0. We are in acompletely analogous situation as the previous case, so by § t = s . • The cone σ x ,y ,z gives the isolated singularity (2 , , x ,y ,z on Y . Inthis chart we get the equation z − y x + s x + s y = 0, which is quasi-smooth because there is no constant term and z appears with degree 1.So all fibres of X → A have a (1 ,
2) singularity at the 0-stratum of thischart of Y . • The cone σ x ,y ,z gives the isolated singularity (1 , , x ,y ,z on Y . Theequation is z − y x + s y + s x = 0 and, in a way analogous to theprevious case, we get a (1 ,
2) singularity on every fibre of X → A at the0-stratum of this chart of Y . • The cone σ x ,y ,z gives the non-isolated singularity (1 , , x ,y ,z . Theequation is z − y x + s x y + s = 0. Since it is quasi-smooth, this givesa (1 ,
1) singularity on every fibre of X → A at a point on the curve( x = y = 0) ⊂ Y . • The cone σ x ,y ,z gives the non-isolated singularity (1 , , x ,y ,z . Theequation is z − y x + s + s x y = 0 and, similarly to the previous case,we have a (1 ,
1) singularity on every fibre of X → A at a point on thecurve ( x = y = 0) ⊂ Y . • In the fan Σ S there are two 3-dimensional cones which we have not beenanalysed yet: these are σ x ,x ,y , whose corresponding chart on Y is the non-isolated singularity (3 , , x ,x ,y , and σ x ,x ,y , which gives the non-isolated singularity (4 , , x ,x ,y on Y . We want to show that it isuseless to analyse these cones. Let V denote the complement in Y ofthe union of the already analysed charts; V is made up of 3 torus-orbits:the 0-stratum corresponding to σ x ,x ,y , the 0-stratum corresponding to σ x ,x ,y , and the 1-stratum corresponding to σ x ,x . In other words V isthe curve ( x = x = 0) in Y . From (5) it is clear that V does not intersectany fibre of X → A .To sum up, we have that the family X → A realises the Q -Gorenstein smooth-ings of the two (1 ,
1) points on X and leaves the (1 ,
1) points and (1 ,
2) pointsuntouched. By versality the family X → A induces a morphism Spf( C [[ s , s ]]) → Def qG ( X ), which is associated to the identification C [[ s , s ]] = C [[ t , t ]], where s = t and s = t . In other words, the base change of X → A to C [[ s , s ]] isthe miniversal Q -Gorenstein deformation of X .2.5. Proof of Theorem 1.1.
Let M (resp. M ) be the connected component of M Kss2 , (resp. M Kps2 , ) containing the surface X . The analysis of the deformationtheory of X in § M is a smooth projective curve.Let X → A be the Q -Gorenstein family considered in § U of the origin in A such that the fibred product X × A U → U inducesa morphism U → M , which is formally smooth at the origin.By looking at the action of Aut( X ) on the base of the miniversal Q -Gorensteindeformation of X , we see that there are K-polystable surfaces in U different from X .Therefore, by composing U → M with M → M , we get a non-constant morphism U → M . By restricting to a general line passing through the origin, we get that M is unirational. Therefore M is rational by L¨uroth’s theorem. This concludes theproof of Theorem 1.1. 3. Mirror symmetry
In [1] some conjectures for del Pezzo surfaces were formulated. In this sectionwe sketch some evidence for these conjectures in the case of the toric del Pezzosurface X and of its Q -Gorenstein deformations. In addition to [1], we refer thereader to [11, 12, 28] and to the references therein for more details about the notionsintroduced below.3.1. Combinatorial avatars of connected components of moduli of delPezzo surfaces.
According to [1, Conjecture A] there is a 1-to-1 correspondencebetween • connected components of the moduli stack of del Pezzo surfaces (with atoric degeneration) and • mutation equivalence classes of Fano polygons. ANDREA PETRACCI c a b b a c Figure 2.
The coefficients of the maximally mutable Laurentpolynomials with Newton polytope P and with T-binomial edgecoefficients (see § Fano polygon is a lattice polygon whose face fan defines a del Pezzo surface(an example is P in § mutation is a certain equivalence relation on Fanopolygons introduced in [2] — we do not give further details here and we refer thereader to [1, 16].The correspondence works in the following way: to (the mutation equivalenceclass of) the Fano polygon P one associates the connected component M of themoduli stack of del Pezzo surfaces which contains the surface X , which is the toricdel Pezzo surface associated to the face fan of P . One has that M is smoothand contains M (the connected component of the K-moduli stack parametrisingK-semistable del Pezzo surfaces and containing X ) as an open substack.3.2. Classical period.
Consider the family of maximally mutable
Laurent poly-nomials with Newton polytope P and with T-binomial edge coefficients [1, Defini-tion 4]. This is the 6-dimensional family f = x y + x − y − + ( x + 2 + x − )( y + y − )+ a xy + a x − y − + b x + b x − + c xy − + c x − y in Q [ a , a , b , b , c , c ][ x ± , y ± ], where a , a , b , b , c , c are indeterminates. InFigure 2 the coefficients of f are written next to the corresponding lattice points of P . The classical period of f is the power series π f ( t ) = (cid:18) π i (cid:19) (cid:90) { ( x,y ) ∈ ( C ∗ ) || x | = | y | = ε } − tf ( x, y ) d xx d yy in Q [ a , a , b , b , c , c ][[ t ]], for some 0 < ε (cid:28)
1. The first coefficients of π f are: π f ( t ) = 1 + 2( a a + b b + c c + 7) t ++ 6( a b + 2 a c + a b + 2 a c + 4 b + 4 b + c + c ) t + · · · . Quantum period.
Let X (cid:48) be the surface corresponding to a general point in M ; in other words, X (cid:48) is a general Q -Gorenstein deformation of the toric surface X .The quantum period of X (cid:48) [26, Definition 3.2] is a certain generating function forgenus zero Gromov–Witten invariants of X (cid:48) which depends on certain parametersrelated to the singularities of X (cid:48) . In this case there are 6 parameters because thesingular locus of X (cid:48) is made up of 2 points of type (1 ,
1) and 2 points of type (1 , X (cid:48) is a hypersurface in the toric Fano Y , one can use the quantum Lefschetztheorem to compute a specialisation of the quantum period of X (cid:48) , i.e. the powerseries G X (cid:48) ∈ Q [[ t ]] obtained from the quantum period by setting the parametersequal to some numbers. This can be done as follows. We use the notation as in § Y is spanned by the divisor classes L + 3 L + 3 L , L + 9 L + 9 L , L + 9 L + 15 L , L + 15 L + 9 L . We consider the cone Λ ⊆ R defined by the inequalities l + 3 l + 3 l ≥ , l + 9 l + 9 l ≥ , l + 9 l + 15 l ≥ , l + 15 l + 9 l ≥ l ≥ ,l ≥ ,l + 3 l + l ≥ ,l + l + 3 l ≥ ,l + 6 l ≥ ,l + 6 l ≥ . The first inequalities say that we are taking (the closure of) the cone of the effectivecurves in N ( Y ) R ; with the second inequalities we are taking the curve classes onwhich the prime torus-invariant divisors of Y have non negative degrees.By using methods similar to [26], one can prove that a specialisation of thequantum period of X (cid:48) is the power series G X (cid:48) ( t ) ∈ Q [[ t ]] equal to (cid:88) ( l ,l ,l ) ∈ Λ ∩ Z (2 l + 6 l + 6 l )! l ! l ! ( l + 3 l + l )! ( l + l + 3 l )! ( l + 6 l )! ( l + 6 l )! t l +5 l +5 l . Notice the following numerology: at the denominator there are the factorial of thedegrees of the prime torus-invariant divisors of Y , the numerator is the factorial ofthe degree of the Q -line bundle O Y ( X (cid:48) ) = 2 L + 6 L + 6 L , the exponent of t isthe degree of the Q -line bundle − K Y − X (cid:48) = 2 L + 5 L + 5 L , which by adjunctionrestricts to − K X (cid:48) . If (cid:80) d ≥ C d t d is the quantum period of X (cid:48) , then the regularised quantum period of X (cid:48) is (cid:80) d ≥ d ! C d t d . From the computation above one computes the first coefficientsof a specialisation of the regularised quantum period of X (cid:48) : (cid:98) G X (cid:48) ( t ) = 1 + 16 t + 936 t + 520 t + 76840 t + 131880 t + 7360920 t ++ 22806000 t + 770459256 t + 3451657440 t + 85553394696 t + · · · . Equality of periods.
A second mirror-symmetric expectation [1, Conjec-ture B] is that there is an equality between • the regularised quantum period of a general surface X (cid:48) in M and • the classical period of the family of maximally mutable Laurent polynomialswith Newton polytope P and with T-binomial edge coefficients.Notice that in our case both periods depend on 6 parameters which should beidentified.Combining § § X (cid:48) and the classical period of the Laurentpolynomial obtained from f by setting a = a = 1 and b = b = c = c = 0: (cid:98) G X (cid:48) ( t ) = π f ( t ) | a = a =1 , b = b = c = c =0 . References [1] M. Akhtar, T. Coates, A. Corti, L. Heuberger, A. Kasprzyk, A. Oneto, A. Petracci, T. Prince,and K. Tveiten,
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Institut f¨ur Mathematik, Freie Universit¨at Berlin, Arnimallee 3, Berlin 14195, Ger-many
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