On the classification of Togliatti systems

In [MeMR], Mezzetti and Mir\'{o}-Roig proved that the minimal number of generators $\mu (I)$ of a minimal (smooth) monomial Togliatti system $I\subset k[x_{0},\dotsc,x_{n}]$ satisfies $2n+1\le \mu(I)\le \binom{n+d-1}{n-1}$ and they classify all smooth minimal monomial Togliatti systems $I\subset k[x_{0},\dotsc,x_{n}]$ with $2n+1\le \mu(I)\le 2n+2$. In this paper, we address the first open case. We classify all smooth monomial Togliatti systems $I\subset k[x_{0},\dotsc,x_{n}]$ of forms of degree $d\ge 4$ with $\mu(I)=2n+3$ and $n\ge 2$ and all monomial Togliatti systems $I\subset k[x_0,x_1,x_2]$ of forms of degree $d\ge 6$ with $\mu(I)=7$.


Introduction
The study and classi cation of smooth projective varieties satisfying at least one Laplace equation is a long standing problem in algebraic geometry. In [6], shedding new light on this subject, it was related to another long standing problem in commutative algebra: the study and classi cation of homogeneous artinian ideals failing the weak Lefschetz property (WLP). We contribute to these two problems resolving the rst question that was le open in [4].
To be more precise. Let k be an algebraically closed eld of characteristic 0, R = k[x 0 , . . . , x n ] and I = (F 1 , . . . , F s ) ⊂ R a homogeneous artinian ideal generated by forms of the same degree d. Set A = R/I. We say that A fails the WLP from degree d − 1 to degree d if the homomorphism ×ℓ : [A] d−1 → [A] d induced by a general linear form ℓ has not maximal rank. As shown in [6], if s ≤ n+d−1 d then this assertion is equivalent to saying that the projection X I n,d of the dth Veronese variety V(n, d) ⊂ P n from F 1 , . . . , F s satis es at least one Laplace equation of order d − 1. We call Togliatti systems the ideals satisfying these two equivalent statements (see De nition 2.3). The name is in honour of Togliatti who gave a complete classi cation of rational surfaces parameterized by cubics and satisfying at least one Laplace equation of order 2 (see for instance [12,13]). Narrowing the eld of study we deal only with monomial ideals I, so X I n,d turns out to be a toric variety. In this sense, one can apply pure combinatoric tools due to Perkinson in [10] to see whether I is a minimal monomial (smooth) Togliatti system (see De nition 2.3 and Propositions 3.4 and 3.6). In [6], using these tools, Mezzetti, Miró-Roig and Ottaviani classi ed all smooth minimal monomial Togliatti systems of cubics in four variables and conjectured a further classi cation for n ≥ 3. By means of graph theory, this conjecture was proved by Miró-Roig and Michałek [7] where a classi cation of smooth minimal Togliatti systems I ⊂ k[x 0 , . . . , x n ] of quadrics and cubics is achieved. When d ≥ 4 the picture becomes much more involved and a complete classi cation seems out of reach for now. Therefore, in [4] it was introduced another strategy: First to establish lower and upper bounds, depending on n and d ≥ 2, for the minimal number of generators of a monomial Togliatti system and then to study the monomial Togliatti systems with xed number of generators.
In fact, in [4], Mezzetti and Miró-Roig bounded the number of generators of monomial Togliatti systems and classi ed all minimal monomial (smooth) Togliatti systems reaching the lower bound or close to reach it; namely those generated by 2n + 1 and 2n + 2 forms of degree d ≥ 4. In this paper, we use again combinatoric tools and we classify the rst open case, i.e., all minimal monomial Togliatti systems generated by 2n + 3 forms of degree d ≥ 4 and n ≥ 2.
Next we outline the structure of this note. In Section 2 we x the notation and we collect the basic results needed in the sequel. Then, in Section 3 we expose the main results of this note. First, we give a complete classi cation of all minimal monomial Togliatti systems generated by 7 forms of degree d ≥ 10 in three variables (see Theorem 3.8). In order to achieve this classi cation we have had to consider all possible con gurations of these 7 monomials regarded in the integer standard simplex d 2 ⊂ Z 3 and then apply combinatorial criteria to each con guration. Separating the problem in two basic cases which we have also had to separate into a few more subcases has helped so as to reduce the number of con gurations to study. Once seen this classi cation we compute all minimal monomial Togliatti systems generated by 7 forms of degree 6 ≤ d ≤ 9 getting a complete scene of what occurs in three variables. From this result we can look apart all minimal monomial smooth Togliatti systems in three variables generated by 7 forms of degree d ≥ 6 and close the open question we were dealing with.

Preliminaries
We x k an algebraically closed eld of characteristic zero, R = k[x 0 , . . . , x n ] and P n = Proj(k[x 0 , . . . , x n ]). Given a homogeneous artinian ideal I ⊂ k[x 0 , . . . , x n ], we denote by I −1 the ideal generated by the Macaulay inverse system of I (see [6,Section 3] for details). In this section we x the notations and the main results that we use throughout this paper. In particular, we quickly recall the relationship between the existence of homogeneous artinian ideals I ⊂ k[x 0 , . . . , x n ] failing the WLP; and the existence of (smooth) projective varieties X ⊂ P N satisfying at least one Laplace equation of order s ≥ 2. For more details, see [6] and [7].
De nition 2.1. Let I ⊂ R be a homogeneous artinian ideal. We say that R/I has the WLP if there is a linear form L ∈ (R/I) 1 such that, for all integers j, the multiplication map has maximal rank, i.e., it is injective or surjective. We o en abuse notation and say that the ideal I has the WLP. If for the general form L ∈ (R/I) 1 and for an integer number j the map ×L has not maximal rank we say that the ideal I fails the WLP in degree j.
Though many algebras are expected to have the WLP, establishing this property is o en rather di cult. For example, it was shown by Stanley [11] and Watanabe [14] that a monomial artinian complete intersection ideal I ⊂ R has the WLP. By semicontinuity, it follows that a general artinian complete intersection ideal I ⊂ R has the WLP but it is open whether every artinian complete intersection of height ≥ 4 over a eld of characteristic zero has the WLP. It is worthwhile to point out that the WLP of an artinian ideal I strongly depends on the characteristic of the ground eld k and, in positive characteristic, there are examples of artinian complete intersection ideals I ⊂ k[x 0 , x 1 , x 2 ] failing the WLP (see, e.g. [9,Remark 7.10]).
In [6], Mezzetti et al. showed that the failure of the WLP can be used to construct (smooth) varieties satisfying at least one Laplace equation of order s ≥ 2 (see also [4,5,7]). We have: Theorem 2.2. Let I ⊂ R be an artinian ideal generated by r forms F 1 , . . . , F r of degree d with r ≤ n+d−1 n−1 . The following conditions are equivalent: (1) the ideal I fails the WLP in degree d − 1; (2) the homogeneous forms F 1 , . . . , F r become k-linearly dependent on a general hyperplane H of P n ; (3) the closure X := Im(ϕ (I −1 ) d ) ⊂ P ( n+d d )−r−1 of the image of the rational map ϕ (I −1 ) d : P n P ( n+d d )−r−1 associated to (I −1 ) d satis es at least one Laplace equation of order d − 1.
The above result motivates the following de nition: De nition 2.3. Let I ⊂ R be an artinian ideal generated by r forms F 1 , . . . , F r of degree d, r ≤ n+d−1 n−1 . We say: (i) I is a Togliatti system if it satis es the three equivalent conditions in Theorem 2.2.
(ii) I is a monomial Togliatti system if, in addition, I (and hence I −1 ) can be generated by monomials.
(iii) I is a smooth Togliatti system if, in addition, the n-dimensional variety X is smooth.
(iv) A monomial Togliatti system I is minimal if I is generated by monomials m 1 , . . . , m r and there is no proper subset m i 1 , . . . , m i r−1 de ning a monomial Togliatti system.
The names are in honor of Eugenio Togliatti who proved that for n = 2 the only smooth Togliatti system of cubics is [12,13]). This result has been reproved recently by Brenner and Kaid [1] in the context of WLP. Indeed, Togliatti gave a classi cation of rational surfaces parameterized by cubics and satisfying at least one Laplace equation of order 2: There is only one rational surface in P 5 parameterized by cubics and satisfying a Laplace equation of order 2; it is obtained from the 3rd Veronese embedding V(2, 3) of P 2 by a suitable projection from four points on it. In [6], the rst author together with Mezzetti and Ottaviani classi ed all smooth rational 3-folds parameterized by cubics and satisfying a Laplace equation of order 2, and gave a conjecture to extend this result to varieties of higher dimension. This conjecture has been recently proved in [7]. Indeed, the rst author together with Michałek classi ed all smooth minimal Togliatti systems of quadrics and cubics. For d ≥ 4, the picture becomes soon much more involved than in the case of quadrics and cubics, and for the moment a complete classi cation appears out of reach unless we introduce other invariants as, for example, the number of generators of I.

The classi cation of Togliatti systems with 2n + 3 generators
From now on, we restrict our attention to monomial artinian ideals I ⊂ k[x 0 , . . . , x n ], n ≥ 2, generated by forms of degree d ≥ 4. It is worthwhile to recall that for monomial artinian ideals to test the WLP there is no need to consider a general linear form. In fact, we have Let I ⊂ k[x 0 , . . . , x n ] be a minimal monomial Togliatti systems of forms of degree d and denote by µ(I) the minimal number of generators of I. In [8], the rst author and Mezzetti proved: In addition, the Togliatti systems with number of generators reaching the lower bound or close to the lower bound were classi ed. Indeed, we have Theorem 3.2. Let I ⊂ k[x 0 , . . . , x n ] be a minimal monomial Togliatti system of forms of degree d ≥ 4. Assume that µ(I) = 2n + 1. Then, up to a permutation of the coordinates, one of the following cases holds: (i) n ≥ 2 and I = (x d 1 , . . . , x d n ) + x d−1 0 (x 0 , . . . , x n ), or (ii) (n, d) = (2, 5) and I = ( . Furthermore, (i) and (ii) are smooth while (iii) is not smooth.
Proof. See [4,Theorem 3.7]. Theorem 3.3. Let I ⊂ k[x 0 , . . . , x n ] be a smooth minimal monomial Togliatti system of forms of degree d ≥ 4. Assume that µ(I) = 2n + 2. Then, up to a permutation of the coordinates, one of the following cases holds: (i) n ≥ 2 and I = ( Proof. See [4, Theorem 3.17 and Proposition 3.19]. In this paper, we address the rst open case and we classify all smooth minimal monomial Togliatti systems I ⊂ k[x 0 , . . . , x n ] of forms of degree d ≥ 4 with µ(I) = 2n + 3 (see Theorem 3.9) as well as all minimal monomial Togliatti systems I ⊂ k[x 0 , x 1 , x 2 ] of forms of degree d ≥ 6 with µ(I) = 7 (see Theorem 3.8).
In order to achieve this classi cation, we associate to any artinian monomial ideal a polytope and we tackle our problem with tools coming from combinatorics. In fact, the failure of the WLP of an artinian monomial ideal I ⊂ k[x 0 , . . . , x n ] can be established by purely combinatoric properties of the associated polytope P I . To state this result we need to x some extra notation.
Given an artinian monomial ideal I ⊂ k[x 0 , . . . , x n ] generated by monomials of degree d and its inverse system I −1 , we denote by n the standard n-dimensional simplex in the lattice Z n+1 , we consider d n and we de ne the polytope P I as the convex hull of the nite subset A I ⊂ Z n+1 corresponding to monomials of degree d in I −1 . As usual we de ne: We have the following criterion which will play an important role in the proof of our main result.
The following criterion allows us to check if a subset A of points in the lattice Z n+1 de nes a smooth toric variety X A or not.
. , x n ] be an artinian monomial ideal generated by r monomials of degree d. Let A I ⊂ Z n+1 be the set of integral points corresponding to monomials in (I −1 ) d , S I the semigroup generated by A I and 0, P I the convex hull of A I and X A I the projective toric variety associated to the polytope P I . Then X A I is smooth if and only if for any non-empty face Ŵ of P I the following conditions hold: As a direct application of this criterion we get: ] be the minimal monomial Togliatti system of cubics described in Example 3.5. Applying the above smoothness criterion we get that the toric variety X A I is smooth.
and we consider the sets of ideals: be a minimal monomial Togliatti system of forms of degree d ≥ 10.
Assume that µ(I) = 7. Then, up to a permutation of the coordinates, one of the following cases holds: Proof. It is easy to check that all of these ideals are minimal Togliatti systems. Vice versa, let us write We consider A I ⊂ d 2 ∩ Z 3 and we slice A I with planes in three possible manners: We divide the proof in two cases: There exists one variable whose square divides all monomials m i .
In both cases, a straightforward computation using the hypothesis d ≥ 10 shows that when we restrict to x 0 + x 1 + x 2 the 7 monomials remain k−linearly independents. Therefore, I is not a Togliatti system.

C
2: Without loss of generality we can suppose that x 2 0 divides each monomial m i . We can also assume that C 2B: u := a 1 > a 2 = a 3 = a 4 =: s ≥ 2.
which is a contradiction. Then, up to permutation of variables, I is as one of the following cases: ).
C b1: In this case we are removing three points from H 0 s and one from H 0 s+1 . Up to permutation of the variables y and z, we can assume If e ≥ 1, then #Ã and we study the four possibilities. If a, c ≥ 2 then we have the factorization which is not a Togliatti system. Otherwise, s = d − 3, then we have several possibilities: . Both of them are not minimal.
If d − s − 1 ≥ a ≥ 2 and s ≤ d − 3. We have e = 0, c = 1 and (m 1 , m 2 , m 3 , In the rst case, we have the factorization For s = d − 3 it corresponds to a Togliatti system of type (4), while for s ≤ d−4 is not Togliatti because when we restrict to x 0 +x 1 +x 2 = 0 the generators, they remain k−linearly independent.
We argue as in the case u = s + 1 to prove that e = 0. Let us consider #(F d−s−1 ∩Ã (1,1) I ). Using the same argumentation we prove that if a, c ≥ 2 we get a contradiction. If a = 1, then either s = d − 3 or s = d − 4 and we have the following cases: . All of them are Togliatti systems of type (4).
The only one which is a minimal Togliatti system is the second one and it is of type (5). Now, we assume e = 0, c = 1 and d − s − 2 ≥ a ≥ 2. In particular, s ≤ d − 4. We consider #(F d−s−1 ∩Ã then a = d − s − 3 (resp. b = d − s − 1). Otherwise we would incur again to a contradiction with the minimality of I. So, we have three possibilities. ( . A er restricting to x 0 + x 1 + x 2 = 0, we see that none of them corresponds to a Togliatti system. To nish with the case b2, we see what happens when d − s − 2 ≥ e ≥ ⌊ d−s−2 2 ⌋. With the same argument that we use before, we can see that a = e. The di erence with the case u = s + 1 is that in this case we can have m 1 and m 2 aligned vertically. This condition can be translated as the case when If this does not happen (i.e. b > a + 2), then it will contradict the minimality of I. Indeed: let us suppose that 0 ≤ k : Therefore it must be b = a + 2 and, since b > c > a we have c = a + 1. Finally we get:

C 2C:
We assume that u := a 1 = a 2 > a 3 = a 4 =: s ≥ 2. Arguing as in case 2B we get u = s + 1 and I and using the minimality assumption, we see that the only two possibilities are either e ≥ 2 and b ≥ 2 or e = b = 1. In the rst case F d−s−2 factorizes as F d−s−2 = L 1 1 F d−s−3 . Repeating the same argument we get that it must be e = b in any case. Now, we considerÃ I as before and we take In the second and third cases we obtain directly a contradiction with the minimality of I. In the  If a ≥ e + 2 we have the factorization F a−e = L 1 e+1 · · · L 1 a−1 F 1 , which contradicts the minimality of I. Therefore, a = e + 1 which is of type (1).

C 2:
We assume e = 0 and b ≥ 1. Let us consider ) and arguing as in the previous cases we get three possibilities: Restricting the generators to the hyperplane x 0 +x 1 +x 2 , we see that each of them is a Togliatti system if, and only if s = d − 3.
Case (iii): assume e = 0, c = 1 and b ≥ 2. In particular a ≥ 3 and s ≤ d − 4. Arguing as before, we see that the only viable possibility is , which is never a Togliatti system.
which is not a Togliatti system.
).  Case (i). Assume that a = 1, then it has to be either s = d − 2 or s = d − 3. Hence I is one of the following possibilities: . And only the rst and the third possibilities give to minimal Togliatti systems of type (1) and type (4) respectively.
Case (ii). Assume that a = 2, then it can be either s = d − 3, s = d − 4 or s = d − 5. Therefore, I is one of the next ideals: . Only the rst and the second ones correspond to minimal Togliatii systems.
Case (iii). Assume that a ≥ 3 which implies that either c = 1 or c = 2 and we have s ≤ d − 4. In both cases, since a ≥ 3 and c ≤ 2, m 1 cannot be aligned vertically with any m i . Therefore, in both cases, we get a contradiction with the minimality of I.
Case (i). We assume b = 0, then considering #(F d−s−1 ∩Ã (1,1) I ) and using the bound for e, we obtain that a = 1 and therefore it is either s = d − 2 or d − 3. So, I is one of the following ideals: . And any of them are minimal Togliatti systems.
Case (ii). We assume b ≥ 1. In this case, we can assume e ≥ b (the other case is symmetric) and we obtain the factorization F d−s−1 = L 1 0 · · · L 1 b−1 F d−s−b−1 . If e > b we get a contradiction with the minimality of I. Hence, e = b. Now we consider as beforeÃ I and we have In the second and third cases we get immediately a contradiction with the minimality. In the rst case, we can repeat the same argument and we get contradiction unless m 1 and m 3 are aligned vertically. Hence, we always obtain that c = a + 1, and we have the factorization which is of type (1).

C 2D:
We assume that u := a 1 > v := a 2 > a 3 = a 4 =: s ≥ 2. Recall that we have the factorization F d−1 = L 0 0 L 0 1 · · · L 0 s−1 F d−s−1 and we easily check that the minimality of I forces v = s + 1. So we ) with u = s + r and d − s − 1 ≥ r ≥ 2. We can assume d − s − 1 ≥ b ≥ ⌊ d−s−1 2 ⌋ and d − s ≥ c > e ≥ 0, and we have d − s − r ≥ a ≥ 0 and s ≤ d − 3. Let us suppose rst that ⌊ d−s−1 2 ⌋ > e ≥ 0. We consider  Case (ii). We assume b = e + 1. Let us considerÃ I as we did before and we examine In the second and third possibilities we obtain directly a contradiction with the minimality of I. In the last possibility we also obtain a contradiction. In fact, if c = d−s and s+r = d−e, we do not remove any Therefore if b = e + 1, it must be d − s − 1 ≥ c ≥ e + 2 and 1 ≤ d − s − e − r. Iterating this argument we conclude that either c = e + 2 and r = 2 or c = e + 3 and r = 3. Therefore, either Case (iii). Arguing as in case (ii) we get b = e + 2 and r = 2. Therefore, Case (iv). Arguing as in case (ii) we get that r = 2 and I is of type (1).
C d2: In this case we assume e = 0 and a ≥ 1. We will separate the case b = 1 from the case b ≥ 2.
. The rst one is not minimal and the remaining two are of type (4).
Case (ii). Assume that a = 1 and c ≥ 2. Suppose that d − s − r − 1 > 0 and let us consider The rst possibility contradicts the minimality of I. Now let us suppose that d − s − r − 1 > 1. In this case, the second (resp. third) possibility can occur if, and only if b = c = d − s − 1 (resp. b = d − s − 2 and c = d − s). Therefore I is one of the next types: which is a Togliatti system if, and only if r = 2 and s = d − 5 (of type (6)).
If d − s − r − 1 = 1, then there are no special restrictions for the second and third case. Therefore I is one of the next types: which is a Togliatti system if, and only if s = d − 4 and c = 3 (of type (5)), or which is a Togliatti system if, and only if s = d − 5 and b = 2 (of type (6)), or which is a Togliatti system if, and only if r = 2 and s = d − 3 (of type (3)).
Now, let us suppose that d − s − r − 1 = 0. Arguing as usual, we see that it cannot be d  (4) and (5) which is a Togliatti system if, and only if s = d − 3 (of type (4)).
Case (iii). Assume c = 1 and a ≥ 2. We consider #(F d−s−1 ∩Ã (0,2) I ) and we obtain that I is one of the next types: which is a Togliatti system if, and only if r = 2 and d − 3 ≥ s ≥ d − 4 (of type (4) and (5)).
which is a Togliatti system if, and only if r = 2 and s = d − 3 (of type (4)).
It is a Togliatti system if, and only if s = d − 4 (of type (5)).
If d − s − r − 1 = 0, we use the same argumentation to prove that d − s − 1 ≥ b ≥ d − s − 2 and then we have two possibilities: ). They do not correspond to a Togliatti system. b = d − s − 1, c = d − s. If d − 2 > s + r, then we have the factorization F d−s−1 = L 2 1 · · · L d−s−r−1 F r and #(F r ∩Ã (0,2) I ) = d − s − 2 > r, which contradicts the minimality of I. Hence we have s + r = d − 2 and I = ( ) which is never a Togliatti system since s ≤ d − 4.
To nish, assume d −s−r = 1. Arguing in the same manner, we see that it cannot occur d −s−3 ≥ b and d − s − 2 ≥ c ≥ 1. Therefore we have the following possibilities: ). It is a Togliatti system if, and only if s = d − 3 and d − s ), it is a Togliatti system if, and only if s = d − 3 (of type (4)). Case (i). We assume b = 1. Since b ≥ ⌊ d−s−1 2 ⌋ it must be s = d − 3. Therefore I = 0 (x 2 0 x 2 , x 0 x 1 x 2 , x 1 x 2 2 , x 3 2 ). The rst one is not minimal while the remaining two are of type (1).
. All of them are of type (4).
. Only the h one is a minimal Togliatti system, and it is of type (5).
Finally, if s = d − 5, I has 15 possibilities, but any of them is a minimal Togliatti system. To nish case 2D we see what happens when d − s ≥ c > e ≥ ⌊ d−s−1 2 ⌋. We see using the minimality that either a ≥ b = e, b ≥ a = e or e ≥ a = b.
Arguing as before we see that in the rst possibility m 1 and m 3 must be vertically aligned and in particular c = e + 2, a = e and r = 2. Therefore I = (x d 0 , x d 1 , x d 2 ) + x s 0 x e 1 x d−s−e−2 2 (x 2 0 , x 0 x 1 , x 2 1 , x 2 2 ) which is of type (1). Now we assume b ≥ a = e. If b, c ≥ e + 1, then we have the factorization F d−s−e−1 = L 1 e+1 F d−s−e−2 and, since #Ã (e,1) I = d − s − e − 1 we get F d−s−e−1 = L 1 e L 1 e+1 F d−s−e−3 which contradicts the minimality. Now, suppose b = e + 1 and c ≥ e + 2 (resp. b ≥ e + 2 and c = e + 1). As we have seen earlier, m 1 and m 3 (resp. m 2 ) must be aligned. Therefore, we can see using the minimality assumption that r = 2 and c = e + 2 (resp. r = 2 and b = e + 2). In both cases I is of type (1).
Finally, let us assume e ≥ a = b. If e ≥ a + 2, we get a contradiction with the minimality of I. Hence either e = a or e = a + 1. If e = a we see that c = a + 2 and r = 2. Therefore I is of type (1). Otherwise e = a + 1 and we get c = a + 2 and r = 2 and I is of type (1).
For any integer d ≥ 3, set M 0 (d) = {x a 0 x b 1 x c 2 | a + b + c = d and a, b, c ≥ 1}.