On Jet Schemes of Pfaffian Ideals 3

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and [GhJoSe], other properties of jet schemes of determinantal varieties are investigated, also from a computational and combinatorial point of view. The reader should keep in mind that, if the ambient space of the original ideal I is N dimensional, that of the kth jet scheme ideal I(k) of I is kN dimensional. Together with the fact that brute force Gröbner bases computations are very expensive, this fact explains why the calculations of determinantal examples are very limited 1. n,k In this note, following the approach of [KoSe], we focus on jet schemes ideals Ir of pfaffian ideals - i.e. ideals generated by all of the pfaffians of size 2r of a skew-symmetric n × n matrix - which is a topic that is is not treated in the literature yet. Observe that, determinantal varieties seldom verify the hypothesis of Mustaţă’s Theorem, e.g. pfaffian varieties are complete intersections only in the trivial cases of 2 pfaffians i.e. variables, and if n is even and the pfaffian ideal I is principal. Specifically, in the first section we recall some basic facts about classic pfaffian ideals; in Section 2 we prove some results which yield n,k n1,k1 a reduction process from the study of Ir to that of some Ir1 , where, n1 ≤ n, k1 ≤ k, r1 ≤ r. In the remaining sections we prove the main results of this paper, cf. Theorems 3.1, 3.6, 4.1 and Corollary 4.10; in particular we establish in which cases the corresponding jet scheme varieties are irreducible.

1. Classical pfaffian Varieties Let K be an algebraically closed ﬁeld and let X be an n × n skew symmetric matrix of indeterminates over K, i.e. the entries xij of X with i < j are indeterminates, xij = −xji for i > j, and xii = 0for i =1,...,n. Let K[X]= K[xij :1 ≤ i < j ≤ n] be the polynomial ring associated with the matrix X. Since a submatrix determined by rows and columns indexes a1,...,a2r of a given skew-symmetric matrix is again skew-symmetric, one lets [a1,...,a2r]

denote the pfaffian of the submatrix (xaiaj ) of X, with i, j = 1,..., 2r ≤ n, which is called a 2r-pfaffian or a pfaffian of size 2r. Let Ir = Ir(X) be the ideal of K[X] generated by all the 2r-pfaffians of X, which is the classical pfaffian ideal and Pr the corresponding pfaffian n(n−1)/2 variety in A . Its coordinate ring Rr(X) = K[X]/Ir(X) has been studied by many authors, cf. for instance [AD], [Av], [De], [GhKr], [JóPr], [KlLa], [Ma]; we collect in the following proposition some of its most important properties.

Proposition 1.1. The ring Rr(X) is a Gorenstein normal factorial domain of dimension (r − 1)(2n − 2r + 1). The a-invariant of Rr(X) is (r − 1)n and its multiplicity is 2n − 4r +2 2n − 4r +2 e(R (X)) = det − . r n − 2r − i + j +1 n − 2r − i − j +1 i,j=1,...,r−1 Its Hilbert series is given by the formula

r−1 z−( 2 ) det(a (z)) H(R ,z)= ij , r (1 − z)d

1 6,3 For instance, the computations we performed on the quotient ring of I2 with [Macaulay2] break up after a few minutes, whereas with [CoCoA] it exhausts the available 250Gb RAM memory in a few hours time. ON JET SCHEMES OF PFAFFIAN IDEALS 3

where d = dim(Rr(X)), and, for 1 ≤ i, j ≤ r − 1, aij(z) is equal to n − 2r + j n − 2r + i n − 2r + j + i − 1 n − 2r +1 − zk. k k k − 1 k +1 k≥ X0 The results contained in the above proposition show on the one hand that pfaffian rings are endowed with many good properties, and on the other that its basic invariants can be computed despite their complexity. We also recall that, by [HeTr] and [Ku], the generators of Ir(X) form a Gröbner basis with respect to any anti-diagonal order, and that similar results have been proven for larger classes of pfaffian ideals by the first author et al., see [DeGo], [DeGo2], [DeSb]. In the following proposition we recall a standard localization argument for pfaffian ideals (cf. [JóPr, Lemma 1.2]), an adaptation of which is needed for the proof of the Theorem 2.4, which follows from the fact that pfaffian ideals are invariant under elementary operations on rows and columns which preserve skew-symmetry.

Proposition 1.2. Let X = (xij), Z = (zij) be two skew-symmetric matrices of size n × n and (n − 2) × (n − 2) respectively, and let Ψ be the set of the indeterminates contained −1 in the (n − 1)th and nth rows and columns of X. Then the map φ: K[X][xn−1,n] −→ −1 K[Z][Ψ][xn−1,n] deﬁned by

xi,nxj,n−1 − xi,n−1xj,n φ(xij)= xij if xij ∈ Ψ and φ(xij)= zij − otherwise, xn−1,n −1 −1 is a ring isomorphism which yields that Rr(X)[xn−1,n] ≃ Rr−1(Z)[Ψ][xn−1,n].

2. Jet schemes and pfaffian Varieties Let K be an algebraically closed field and let n, k be two fixed positive integers; let n,k (h) also A = K[xij : 1 ≤ i < j ≤ n, h = 0,...,k − 1] be a polynomial ring over K, of dimension of kn(n − 1)/2. Now let t be an indeterminate over K, and X(t)=(xij(t)) be (0) (1) (k−1) k−1 the n × n skew-symmetric matrix with entries xij(t) = xij + xij t + . . . + xij t , for i < j. Let r ∈ N+, such that 2r ≤ n. The main object of our study are the so called jet n,k n,k n,k n,k n,k scheme ideals (or jet ideals for short) Ir ⊆ A , and the quotient rings Rr = A /Ir , which we are going to define as follows. Any pfaffian p = p(t) of X(t) can be written as p = p(0) + p(1)t + . . . + p(k−1)tk−1 + pktk + . . . + p(s)ts, for some integer s and p(h) ∈ An,k, n,k (h) h =1,...,s. The ideal Ir is defined as the ideal generated by p , for all h =0,...,k − 1 and all 2r-pfaffians p of X(t). Example 2.1. Let n =5, r =2, k = 3 and X(t) be the skew-symmetric matrix

0 x12(t) x13(t) x14(t) x15(t) 0 x (t) x (t) x (t)  23 24 25  X(t)= 0 x34(t) x35(t) ;  0 x45(t)    0     5,3 (h)  we are interested in ideals such as I2 = (pi : h = 0, 1, 2, i = 1,..., 5), where pi denotes here the pfaﬃan of the matrix obtained from X(t) by deleting its i th row and column. For 4 EMANUELADENEGRIANDENRICOSBARRA

5,3 instance, p5 = x12(t)x34(t) − x13(t)x24(t)+ x14(t)x23(t) gives rise to the three generators of I2 (0) (0) (0) (0) (0) (0) (0) p5 = x12 x34 − x13 x24 + x14 x23 ,

(1) (0) (1) (0) (1) (0) (1) (1) (0) (1) (0) (1) (0) p5 = x12 x34 − x13 x24 + x14 x23 + x12 x34 − x13 x24 + x14 x23 ,

(2) (0) (2) (0) (2) (0) (2) (1) (1) (1) (1) (1) (1) (2) (0) (2) (0) (2) (0) p5 = x12 x34 − x13 x24 + x14 x23 + x12 x34 − x13 x24 + x14 x23 + x12 x34 − x13 x24 + x14 x23 , n,1 Clearly, when k = 1, the ideal Ir is the classical pfaﬃan ideal Ir of a generic skew-symmetric n,1 matrix of size n generated by all pfaﬃans of size 2r and Rr ≃ Rr is the quotient ring which we described in the previous section. The following lemma and theorem will be relevant for the induction arguments we are going to use later on. (0) (0) Lemma 2.2. Let (x ) ⊂ K[X] denote the ideal generated by xij , for all 1 ≤ i < j ≤ n. n,k−1 n,k (0) R , if k ≤ r; Sr /(x ) ≃ n,k−r (h) ( Sr [xij ]h=k−r+1,...,k−1 , if k > r.

Proof. Let p = [a1,...,a2r]bea2r-pfaﬃan of X(t). By one of the equivalent deﬁnition of

pfaﬃan, we may write p = α∈Π sgn(σα)xi1,j1 (t) ··· xir,jr (t), where i < i <...r. By the same argument, (p(h) : 0 ≤ h ≤ r − 1) ⊆ (x(0)). Now, if r ≤ h ≤ k − 1, then

(h) (l1) (lr ) (l1) (lr ) p = sgn(σα)  xi1,j1 ··· xir,jr + xi1,j1 ··· xir,jr  α∈Π l1+...+lr=h l1+...+lr=h P  li>P0, ∀i li>P0, ∃i    (l1) (lr ) = sgn(σα)  xi1,j1 ··· xir,jr  + g , α∈Π l1+...+lr=h P  li>P0, ∀i    where g ∈ (x(0)). Now we only need to observe that the ﬁrst summand, after relabeling, is n,k−r just a generator of Ir , and the proof is completed. s k Remark 2.3. In general, if A is any ring, f = f0 + f1t + ··· + fst ∈ A[t] and B = A[t]/(t ), then it is easy to see that f is invertible in B if and only if f0 is invertible in A. In our setting we consider jet ideals, which are ideals in the ring A = An,k, but we might as well consider k k (h) them as ideals in B = A[t]/(t ) ≃ (K[t]/(t ))[xij : 1 ≤ i < j ≤ n, h =0,...,k − 1], where the above property comes at hand; in this case we say that we are working modulo tk. ON JET SCHEMES OF PFAFFIAN IDEALS 5

(h) (h) (h) (h) (h) (h) Theorem 2.4. Let Ψ = {x1n , x2n ,...,xn−1,n, x1,n−1, x2,n−1,...,xn−2,n−1 : h = 0,...k − 1}. Then n,k (0) −1 n−2,k (0) −1 Rr [(xn−1,n) ] ≃ Rr−1 [Ψ][(xn−1,n) ] . n,k (0) Moreover, there is a one-to-one correspondence between {p ∈ Min(Ir ): xn−1,n 6∈ p} and the n−2,k n−2,k set Min(Ir−1 ) of minimal primes of Ir−1 which preserves codimension.

Proof. Let Z(t)=(zij(t)) be the (n − 2) × (n − 2) skew-symmetric matrix whose entries (0) (1) (k−1) k−1 (h) are zij(t) = zij + zij t + . . . + zij t , where zij are indeterminates, and consider the assignment

xi,n(t)xj,n−1(t) − xi,n−1(t)xj,n(t) φ(xij(t)) = xij(t) if xij(t) ∈ Ψ, and φ(xij(t)) = zij(t) − , xn−1,n(t) k otherwise. Working modulo t , by Remark 2.3, one has that xn−1,n(t) is invertible since (0) xn−1,n is invertible. Therefore, by arguing as in the proof of [KoSe, Theorem 1.2], we obtain the desired isomorphism and correspondence.

n,k (0) Corollary 2.5. Let p be a minimal prime of Ir ; then, xij ∈ p for some i and j if and only (0) if xij ∈ p for all i and j. n,k Proof. Without loss of generality, we only have to prove that, if p is a minimal prime of Ir (0) (0) that does not contain xn−1,n, then xij 6∈ p for any choice of i < j. The viceversa will then follow, by exchanging the roles of the indexes n−1, n and i, j. Suppose by contradiction that (0) (0) there exist i and j such that xij belongs to p; by the the previous theorem, xij would be (0) −1 n−2,k (0) −1 (0) −1 then mapped into the image q[Ψ][(xn−1,n) ] ⊂ Rr−1 [Ψ][(xn−1,n) ] of p[(xn−1,n) ], where n−2,k (0) (0) (0) (0) (0) (0) (0) −1 q ∈ Min(Ir−1 ). Since the image of xij is zij − (xin xj,n−1 − xi,n−1xjn )(xn−1,n) , it would (0) (0) (0) (0) (0) (0) follow that zi,j xn−1,n − xi,nxj,n−1 + xi,n−1xj,n ∈ q[Ψ]. Now, the coeﬃcients of this expressions involve ±1, which would then belong to the prime ideal q, a contradiction.

n,k V n,k n,k In the following we let Pr = (Ir ) denote the pfaﬃan jet variety of Ir .

n,k (0) n,k Remark 2.6. If we let M = {p ∈ Min (Ir ) : xij 6∈ p for all i, j}, N = {p ∈ Min (Ir ) : (0) xij ∈ p for some i, j}, I = p and J = p, then we may write p∈M p∈N T T n,k n,k Min (Ir )= M ⊔ N, and Ir = I ∩ J. q (0) (0) V Let, as before, (x ) be the ideal generated by all of the xij ; moreover let Z = (I) and n,k V (0) n,k V V Y = Pr ∩ ((x )). Clearly, Pr = (I) ∪ (J) = Y ∪ Z. Notice that Z is not empty and Z 6⊆ Y; moreover, Z is the union of its irreducible components which correspond to (0) the set of minimal primes deﬁning I, i.e. those not containing xn−1,n. By Theorem 2.4, n−2,k there is a bijection between these and the irreducible components of Pr−1 , which preserves codimensions. Finally, by Lemma 2.2, Y is isomorphic to either An(n−1)(k−1)/2 when k ≤ r n,k−r An(n−1)(r−1)/2 or to Pr × when k>r. 6 EMANUELADENEGRIANDENRICOSBARRA

3. The irreducible cases Philosophically speaking, for classical determinantal and pfaffian varieties, the so called maximal cases are usually endowed with better properties. In this section and in the next one we shall focus on the problem of irreducibility of pfaffian jet varieties (or pfaffian jet varieties for short) and it will turn out that this general behaviour is confirmed. In this section we shall consider the special cases when 2r = n and 2r +1= n, and prove that the corresponding pfaffian jet varieties are irreducible. 2r,k n,k n =2r. This is the easiest case: the ideal Ir = In/2 has exactly k generators, which are p(h), h =0,...,k−1, where p = [1,...,n], the pfaffian of the whole matrix X(t). We consider n,k (k−1) (k−1) (k−1) now the degrevlex order ≺ on the monomials of A induced by x1,2 >...>x1,n > x2,3 > (k−1) (k−2) (k−2) (k−2)) (k−2) (0) (0) (0) ...>xn−1,n > x1,2 >...>x1,n > x2,3 >...>xn−1,n >...>x1,2 >...>x1,n > x2,3 >...> (0) xn−1,n. Theorem 3.1. Let n =2r. Then, 2r,k a) the Gröbner basis w.r.t. to ≺ of the pfaffian jet ideal Ir is given by [1,...,n](0),..., [1,...,n](k−1); 2r,k b) the jet pfaffian ideal Ir is prime of codimension k;

2r,k k c) Rr is a complete intersection and, therefore, Cohen-Macaulay of multiplicity r . Proof. Let p = [1,...,n]. For the proof of a) it is suﬃcient to observe that the leading term (h) (l1) (lr ) of p = α∈Π sgn(σα) l1+···+lr=h xi1,j1 ··· xir,jr is given by (hP) (q) P (q) (q) (q+1) (q+1) (q+1) in≺(p )= x1,2r−2t ··· xr−t−1,r−t+2 · xr−t,r−t+1 · x2r−2t+1,2r · x2r−2t+2,2r−1 ··· x2r−t,2r−t+1 , where h = qr + t, with 0 ≤ t ≤ r − 1. Therefore, it is easy to see that any two distinct such monomials are relatively prime, and it is well known that in this situation one has a Gröbner basis. 2r,k b) : We shall prove that Rr is a domain of dimension d := k(n − 2)(n +1)/2 by induction 2,k h 2,k on r. If r = 1 and n = 2, then I1 =(x12 : h =0,...,k − 1) and R1 ≃ K is a 0-dimensional n−2,k domain. We now let r > 1 and assume that Rr−1 is a domain of dimension k(n−4)(n−1)/2; n−2,k let also Ψ be as in Theorem 2.4. Then, Rr−1 [Ψ] has dimension k(n − 4)(n − 1)/2+ k(n − n−2,k (0) −1 1+ n − 2) = d, which is also the dimension of Rr−1 [Ψ][(xn−1,n) ]. Theorem 2.4 now n,k (0) −1 n,k (0) yields that Rr [(xn−1,n) ] is a domain and so is Rr : in fact, by Part a), xn−1,n is regular 2r,k modulo in≺ Ir and, by a well known property of the reverse lexicographic order, it is also 2r,k n,k n,k (0) −1 Rr -regular. Finally, dim Rr = dim Rr [(xn−1,n) ] = d, as desired. Part c) follows now easily from the above. Remark 3.2. If n = 2r, the kth jet scheme is generated by a complete intersection of k n,k elements of degree r, for any k; therefore we also know the Hilbert series of Rr , which is k r−1 i r−1 i k n,k j=1 i=1 z i=1 z H(R , z)= dim An,k = dim An,1 . (1Q− zP) (1 −Pz) ! ON JET SCHEMES OF PFAFFIAN IDEALS 7

Thus, (3.3) H(Rn,k, z)= H(Rn,1, z)k and, consequently

(3.4) dim Rn,k = k dim Rn,1, e(Rn,k)= e(Rn,1)k.

n =2r + 1. We recall first some general basic facts and set some notation useful henceforth. Let M be any m × m skew-symmetric matrix. For for l =1,...,m we denote by Ml the (m−1)×(m−1) skew-symmetric matrix obtained by removing the lth row and column from M. Accordingly, we denote by pfl(M) the pfaffian of Ml. Let λ ∈ K; if λ =6 0, we let Ei(λ) be the elementary row multiplication matrix and, for any λ, we let Eij(λ) be an elementary ′ ′′ t row addition matrix. If we let M = Ei(λ) M Ei(λ) and M = Eij(λ) M Eij(λ) , then it is well-known that pf M ′ = |λ|pf M and pf M ′′ = pf M. P A(2r+1)rk Let now M be a skew-symmetric matrix of size 2r +1. We associate a point ∈ K (h) of coordinates pij , where 1 ≤ i < j ≤ 2r +1, 0 ≤ h ≤ k − 1, with the matrix AP = (0) (1) (k−1) k−1 P 2r+1,k (pij(t)), where pij(t) = pij + pij t + ··· + pij t , and viceversa. Clearly, ∈ Pr (0) (k−1) if and only if the first k coefficients pfl(AP ) ,..., pfl(AP ) of pfl(AP ) are zero for all l =1,..., 2r + 1.

n,k Lemma 3.5. Let Y, Z ⊂ Pr be as in Remark 2.6, and for any i, j, let Uij be the open set n,k (0) of Pr where xij =06 . Then one has:

a) Z = Uij, where denotes the Zariski closure; b) if n =2r +1, then Y ⊆ Z. Proof. Part a) can be proved along the lines of the proof of [KoSe, Lemma 2.7]. For b), we prove that for all Q ∈ Y there exist i, j, with i < j such that Q ∈ Uij; in this way the conclusion will be yielded by Part a). (0) (1) (k−1) k−1 Let AQ = (qij(t)), with qij(t) = qi,j + qi,j t + ··· + qi,j t , be the matrix associated Q Q 2r+1,k (0) with . Recall that belongs to Y exactly when it belongs to Pr and qij = 0 for all i, j. If Q is the origin, we choose any i < j and a point P ∈ Uij, since the latter is not empty, and consider the family of points Pλ = λP, with λ ∈ K. Clearly, P0 = Q and Pλ ∈ Uij for all λ =6 0; therefore, by the properties of Zariski topology, Q ∈ Uij. Q If is not the origin, consider i0, j0 such that qi0,j0 (t) =6 0; by exchanging rows and columns if necessary, we may assume without loss of generality that i0 =6 2r + 1 and (s) Q j0 > i0 +1. Let s> 0 be the minimum such that qi0,j0 =6 0. We shall show that ∈ Ui0+1,j0 . s Let E = Ei0+1,i0 (λ/t ) be an elementary row addition matrix; starting with AQ we construct t a family of matrices APλ = E AQ E , depending on a parameter λ ∈ K , which corresponds to a family of points Pλ. Clearly, P0 = Q. Moreover, for all λ =6 0, we have that P (0) (0) (s) (s) λ ∈ Ui0+1,j0 , since (pλ)i0+1,j0 = qi0+1,j0 + λqi0,j0 = λqi0,j0 =6 0. In order to conclude the P 2r+1,k (h) proof now we only need to show that λ ∈ Pr for all λ =6 0, i.e. that, pfl(APλ ) =0 for h = 0,...,k − 1 and l = 1,..., 2r +1. If l = i0 + 1, we remove the column and row we operated on and we obtain (APλ )l =(AQ)l and, therefore, pfl(APλ )=pfl(AQ). Let now l = i0, and B be the matrix obtained by AQ by removing the (i0 + 1)th row and column s and multiplying the i0th row and column by a non-zero factor λ/t ; in this case we have 8 EMANUELADENEGRIANDENRICOSBARRA

s that pfl (APλ ) = pfl (AQ)+pf(B) = pfl (AQ)+ λ/t pfl+1 (AQ). Finally, if l =6 i0, i0 +1 t t then (APλ )l = (E AQ E )l = El (AQ)l (El) and, thus, pfl(APλ ) = pfl(AQ). Now, working k (h) modulo t , since by hypothesis pfl(AQ) = 0 for every l and for h = 0,...,k − 1, the conclusion is straightforward.

The next result generalizes what it happens in the classical case: if k = 1 and n =2r + 1, then the pfaﬃan ideal has codimension 3.

2r+1,k Theorem 3.6. Any pfaﬃan jet variety Pr is irreducible of codimension 3k.

3,k Proof. The proof runs by induction on r. When r = 1 and n = 3, the ideal I1 is generated (h) (h) (h) 3,k A3k by {x12 , x13 , x23 : h =0,...,k − 1} and P1 = {O} ⊂ , which is irreducible of the right 2s−1,k codimension. Let us now suppose that r> 1 and that the varieties Ps−1 , for s ≤ r and all 2r−1,k k, are irreducible of codimension 3k. Thus, the radical of Ir−1 is prime and, by Theorem 2r+1,k (0) 2.4, also the radical of Ir is a prime ideal, that does not contain xn−1,n. By Remark 2.6, it is now suﬃcient to prove that Y ⊆ Z, but that is yielded by Lemma 3.5.

5,3 Example 3.7. Consider I2 , the jet pfaﬃan ideal of the ideal generated by the 4 pfaﬃans of 5,3 a generic 5 × 5 skew symmetric matrix X(t) of polynomials of degree 2 in t; I2 is generated 5,3 by 15 polynomials of degree r=2. With Macaulay2 we can verify that codim I2 = 9, as 5,3 5,3 implied by the previous result, and that I2 is prime. Moreover, the Hilbert series of R2 is 1+9z + 30z2 + 45z3 + 30z4 +9z5 + z6 H(R5,3,z)= , 2 (1 − z)21 and 1+3z + z2 H(R5,1,z)= . 2 (1 − z)7

5,3 5,1 3 Thus, H(R2 , z)=(H(R2 , z)) . Some computational evidence suggests that the given set of generators is not a Gröbner basis with respect to monomial orders which are in a natural way mutated from those which are classically considered in the literature for ideals of Pfaﬃans.

The corresponding classical pfaﬃan ideals in this case are even prime; we expect the same 2r+1,k kind of behaviour for the general jet pfaﬃan ideal Ir .

2r+1,k Conjecture 3.8. For all positive integers r and k, the jet pfaffian ideals Ir are prime. Considering the size of the computations involved, an affirmative answer to the following would provide a great simplification for the calculation of the Hilbert series of such rings.

2r+1,k Question 3.9. Do Formulas (3.3) and (3.4) hold true for Rr , for all positive integers r and k?

2r+1,k Question 3.10. Are Pr Cohen Macaulay, for all positive integers r and k? ON JET SCHEMES OF PFAFFIAN IDEALS 9

4. The reducible cases We now want to prove that the pfaﬃan jet ideals are reducible except for the cases considered in the previous section. n ≥ 2r +2 The main result of the section is the following theorem.

n,k Theorem 4.1. The pfaffian jet variety Pr is reducible for every n ≥ 2r +2, with r ≥ 2. n,k Moreover, Pr is not pure; in particular, it is not Cohen Macaulay. In order to prove the theorem, we want to use the reduction argument of Lemma 2.4; therefore we first focus on the case of jet varieties of 4-pfaffians, with n ≥ 6. n,k We start by describing a procedure to obtain the irreducible components of P2 , with n ≥ 6. n,k Recall that, as discussed in Remark 2.6, we may write P2 = Y0 ∪ Z0, where Y0 is the (0) subvariety where xij = 0 for all i, j, and Z0 is the closure of any of the open sets Uij - see Lemma 3.5 a). In contrast to what happens in the maximal case, cf. Lemma 3.5 b), we prove next that there is not an inclusion between the two subvarieties.

n,k Lemma 4.2. Let r =2, n ≥ 6, and P2 = Y0 ∪ Z0. Then, one has Y0 6⊆ Z0.

Proof. Let Q be a point which is in Uij ⊂ Z0 for some i, j, and let AQ = (qij(t)), with (0) (1) (k−1) k−1 Q Q qij(t) = qi,j + qi,j t + ··· + qi,j t , be the matrix associated with . Since belongs n,k (h) to P2 , for every 4-pfaffian p of AQ one has p = 0, for all h = 0,...,k − 1. Let q be any 6 pfaffian of AQ. For what we are going to say next, we may assume without loss of generality that Q is the submatrix of the first 6 rows and columns of AQ, and that q = pf(Q). Since by Laplacian expansion q can be expressed in terms of 4-pfaffians, we have that q(h) = 0, for h = 0,...,k − 1 and, therefore, q = 0 modulo tk. Thus also det Q = 0 modulo tk. Again without loss of generality, we may assume (i, j)=(5, 6) and that, working modulo tk, we may express the last column of Q as a linear combination of the previous ones. Thus, there exist polynomials λh(t), h = 1,..., 5 of degree at most k − 1 such that 5 k qi6(t) = h=1 λh(t)qih(t) mod t for i = 1,..., 5, i.e. there exists polynomials µi such that 5 k qi6(t) = P h=1 λh(t)qih(t)+ t µi for i = 1,..., 5. Consider now the 6 × 6 skew-symmetric matrices Y =(y ), Z =(z ), defined by P ij ij

qij (t) if1 ≤ i < j ≤ 5, qij (t) if1 ≤ i < j ≤ 5, yij = zij =  5 k λh(t)qih(t) if1 ≤ i < j = 6; ( t µ if 1 ≤ i < j =6.  h=1 i P Thus, pf(Y ) = 0 and

5 5 k k (4.3) q = pf(X) =pf(Y )+pf(Z)= ±t µh pfh6 Z = t ±µh pfh6 Q. hX=1 hX=1

From the above, we derive that a point belongs to the (non-empty) open subset U56 ⊂ Z veriﬁes the extra conditions

q(h) =0 for h = k,... 2k − 1. 10 EMANUELADENEGRIANDENRICOSBARRA

On the other hand, consider the point P determined by the associated matrix 0 tℓ 0 0 0 0 0 tℓ 000 0 00 0 0 0 0000 0  0 tℓ+1 0 0   0 tℓ 0 0  (4.4) AP = , if k =2ℓ +1 or AP = ,  0 0 0   00 0       0 tk−1  0 tk−1      0   0       2k−1   if k = 2l; in both cases pf(AP ) = t , and it is not difficult to see that P ∈ Y0 \ Z0 for s =0,...,ℓ − 1. n,k Proposition 4.5. The jet variety P2 of a pfaffian ideal generated by 4-pfaffians is reducible for all n ≥ 6. n,k Proof. The previous lemma implies that P2 = Y0 ∪Z0 yields a decomposition of our variety as a union of two proper subvarieties. n,k Now we go on decomposing, in order to find the irriducible components of P2 . By Lemma n(n−1) n,1 n(n−1) A 2 A 2 2.2, if k = 2 then Y0 ≃ , and if k = 3, then Y0 ≃ P2 × which is irreducible n,1 since P2 is a classical pfaffian variety - see Proposition 1.1; thus, if k = 2, 3, then Y0 is irreducible. n,k−2 A n(n−1) Let now k ≥ 4, so that Y0 ≃ P2 × 2 ; let Y1 be the subvariety of Y0 of the points (1) n(n−1) A 2 with xij =0 for all i, j; we let Z1 = T1 × , where T1 is isomorphic to the subvariety n,k−2 (1) of P2 which is the closure of the open set where some xij =6 0. Roughly speaking, T1 is n(n−1) n,k−2 A if k =4 the “Z” subvariety of P2 . By Lemma 2.2, Y1 ≃ n,k−4 An(n−1) and, if ( P2 × if k> 4 k = 4, then Y1 is irreducible. By repeating this procedure ℓ − 1 times, where ℓ = ⌊k/2⌋, we obtain subvarieties Ys, for (h) s =1,...,ℓ − 1, defined by xij = 0 for h =0,...,s and for all i, j. For s ≤ l − 2, we have n,k−(2s+2) n(n−1) n(n−1) A(s+1) 2 A(s+1) 2 Ys ≃ P2 × and Zs+1 = Ts+1 × , where Ts+1 is isomorphic n,k−(2s+2) (s+1) to the subvariety of P2 which is the closure of the open set where some xij =6 0. ℓn(n−1) A 2 if k =2ℓ; − − Moreover, Yℓ 1 ≃ n,1 ℓn(n 1) This descends by the very definition  A 2  P2 × if k =2ℓ +1. of the jet equations given by 4-pfaffians. In fact, if k = 2ℓ, the defining equations of Yℓ−1 (h) simply become xij = 0for h =0,...,ℓ−1, and the remaining variables are free; if k =2ℓ+1, (h) the equations defining Yℓ−1 do not involve the variables xij for h = ℓ +1,...,k, which are therefore free in the corresponding coordinate ring as in the previous case. Furthermore, n,1 Yℓ−1 is defined by the equations of the classical pfaffian variety P2 expressed in terms of (l) the variables xij . Remark 4.6. With the above notation, we can immediately generalize Lemma 4.2: for s =0,...,ℓ − 1, one has Ys 6⊆ Z0. To this end, one only needs to observe that the point P defined in (4.4) belongs to Ys \ Z0, for all s =0,...,ℓ − 1. As a first consequence of the above discussion, we obtain the following result. ON JET SCHEMES OF PFAFFIAN IDEALS 11

Proposition 4.7. Let ℓ = ⌊k/2⌋. With the above notation, n,k a) For all s =0,...,ℓ − 1, the subvarieties Zs ⊆ P2 are irreducible of codimension (k−2s)(n−2)(n−3) n(n−1) 2 + s 2 .

b) the subvariety Yℓ−1 is irreducible with ℓn(n−1) 2 , if k =2ℓ; codim Yℓ−1 =  − − −  ℓn(n 1) (n 2)(n 3) 2 + 2 , if k =2ℓ +1.  Proof. Recall that, from the above discussion, for all s = 0,...,l − 1, one has Zs = n(n−1) n,k−2s As 2 Ts × , where Ts is isomorphic to the subvariety of P2 which is the closure of the (s) open set where some xij =6 0. It follows from Theorem 2.4 that the irreducible components n−2,k−2s of Ts are in correspondence with the components of P1 , and this correspondence pre- n−2,k−2s (k−2s)(n−2)(n−3) serves the codimensions. Thus, since P1 is irreducible of codimension 2 , (k−2s)(n−2)(n−3) n(n−1) then also Zs is irreducible and codim Zs = 2 + s 2 , and this proves the ﬁrst part of the statement. For Part b), we have already determined Yl−1 and it is immediate to check that it is irreducible. Moreover, it is easy to compute its codimension: if k = 2ℓ, then codim Yℓ−1 = ℓn(n−1) n,1 n(n−1) 2 ; by Proposition 1.1, the codimension of P2 is 2 − 2n + 3 and therefore, if n(n−1) ℓn(n−1) (n−2)(n−3) ℓn(n−1) k =2ℓ + 1, the codimension of Yℓ−1 is − 2n +3+ = + . 2 2 2 2 Lemma 4.8. Let r =2 and n ≥ 6.

a) If n =6, 7, then codim Z0 < codim Z1 <...< codim Zℓ−1 < codim Yℓ−1. n,k Moreover, Z0 is the irreducible component of P2 of smallest codimension and n,k codim P2 = codim Z0.

b) If n ≥ 8, then codim Yℓ−1 < codim Zℓ−1 <...< codim Z1 < codim Z0. n,k Moreover, Yℓ−1 is the irreducible component of P2 of smallest codimension and n,k codim P2 = codim Yℓ−1.

Proof. It is clear by deﬁnition that, for every s, Zs cannot be contained in Zs+1, since (s) xij = 0 in Zs+1 but not in Zs; now, a direct computation shows that the codimension of Zs is a strictly decreasing function in s if and only if n2 − 9n + 12 > 0, i.e for all n ≥ 8; it is strictly increasing for n =6, 7. Proposition 4.9. Let n ≥ 8 and, for any integer k ≥ 2, let ℓ = ⌊k/2⌋. Then, the decompo- n,k n,k sition of P2 into irreducible components is given by P2 = Yℓ−1 ∪ Z0 ∪ Z1 ∪···∪Zℓ−1.

Proof. We have already observed that Zs cannot be contained in Zs+1. Since n ≥ 8, by Lemma 4.8, Zs cannot contain Zs+1. In a similar fashion we may conclude that Zs cannot be contained in Yℓ−1 and cannot contain Yℓ−1; now the conclusion is straightforward. n,k Corollary 4.10. Let n ≥ 2r +4 and r ≥ 2. Then Pr has at least ⌊k/2⌋ +1 irreducible components. 12 EMANUELADENEGRIANDENRICOSBARRA

Proof. We argue by induction on r. If r = 2 then n ≥ 8 and the result holds true by Proposition 4.9. Let now r > 2; since n ≥ 2r + 4, then n − 2 ≥ 2(r − 1) + 4; thus by n−2,k induction Pr−1 has at least ⌊k/2⌋ + 1 irreducible components, and Theorem 2.4 yields that n,k these are in bijection with the irreducible components of the subvariety Z0 of Pr ; this is enough to imply what desired. Remark 4.11. The above result is analogous to that of [KoSe, Theorem 6.1], which treats the general situation of non-maximal minors in the generic case. A more precise statement about the number of irreducible components of jet schemes of determinantal varieties is proved in [Do, Theorem A/Corollary 4.13]. At the moment, we cannot provide a counterpart to the latter in the case of pfaﬃans using these techniques. Proof of Theorem 4.1. We argue again by induction on r, the case r = 2 being settled by n−2,k n,k Proposition 4.5. By induction, Pr−1 is reducible and, again by Theorem 2.4, so is Pr . n,k It is easy to check by means of Lemma 4.8, that the irreducible components of P2 have n,k diﬀerent codimensions; hence, P2 is not pure, and thus not Cohen Macaulay. The same n,k conclusion holds for Pr , by arguing on Z0 and applying Theorem 2.4.

Example 4.12. Conjecture 3.8 is false and Question 3.9 has a negative answer for pfaffian n,k 6,2 jet ideal Ir with n ≥ 2r + 2, as the following example shows. Consider I2 , the jet pfaffian ideal generated by all of the 4 pfaffians of a generic 6 × 6 skew symmetric matrix X(t) of 6,2 polynomials of degree 1 in t; I2 is generated by 30 polynomials of degree r = 2. With 6,2 Macaulay2 one verifies that codim I2 = 12, and that the ideal is radical, but not prime. 6,2 Moreover, the Hilbert series of R2 is 1+12z + 48z2 + 75z3 + 45z4 + 15z5 H(R6,2,z)= , 2 (1 − z)18 whereas 1+6z +6z2 + z3 H(R6,1,z)= . 2 (1 − z)9 Thus, 6,2 6,1 6,2 2 6,1 2 dim R2 = 18 = 2 dim R2 , and e(R2 )=196=14 = e(R2 ) 6,2 6,1 2 6,2 6,1 2 but h(R2 ) =(6 h(R2 )) ; therefore H(R2 , z) =6 H(R2 , z) .

6,2 Remark 4.13. Observe that, in the above example, the non trivial component Z0 of P2 6,2 (0) ∞ is deﬁned by the ideal I2 : (x56 ) ; its Hilbert series is 1+12z + 48z2 + 74z3 + 48z4 + 12z5 + z6 . (1 − z)18 As in the analogous case for the principal component of the 2nd jet ideal of the ideal of 2 × 2 minors in a generic matrix (cf. the main result of [GhJoSe], Theorem 18), we have that its Hilbert series is the square of the original one. We also notice that its h-vector has only positive coeﬃcients and it is symmetric. ON JET SCHEMES OF PFAFFIAN IDEALS 13

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Emanuela De Negri - Dipartimento di Matematica - Università di Genova - Via Dodecaneso 35 - 16146 Genova - Italy - OrcidID 0000-0001-5556-440X - Corresponding author E-mail address: [email protected] Enrico Sbarra - Dipartimento di Matematica - Università di Pisa - Largo Bruno Pon- tecorvo 5 - 56127 Pisa - Italy - OrcidID 0000-0001-9656-6153 E-mail address: [email protected]