Hessian Ideals of a Homogeneous Polynomial

and Generalized Tjurina Algebras

Alexandru Dimca ^{1}, Gabriel Sticlaru

Received: December 8, 2014 Revised: March 21, 2015

Communicated by Thomas Peternell

Abstract. Using the minors in Hessian matrices, we introduce new graded algebras associated to a homogeneous polynomial. When the associated projective hypersurface has isolated singularities, these algebras are related to some new local algebras associated to isolated hypersurface singularities, which generalize their Tjurina algebras. One consequence of our results is a new very rapid way to determine the number of weighted homogeneous singularities of such a hypersurface.

2010 Mathematics Subject Classification: Primary 14J70, 13D40; Secondary 14B05, 32S05 Keywords and Phrases: projective hypersurfaces, graded algebra, Hessian matrix, weighted homogeneous singularities

1. Introduction and statement of results

Let S = ⊕_{k}S_{k} = ℂ[x_{0}, ..., x_{n}] be the graded polynomial ring in n+1 indeterminates with complex coefficients, where S_{k} denotes the vector space of degree k homogeneous polynomials. Consider for a polynomial f ∈ S_{d}, the corresponding Jacobian ideal J_{f} generated by the partial derivatives f_{j} of f with respect to x_{j} for j = 0, ..., n, the graded Milnor algebra M(f) = ⊕_{k}M(f)_{k} = S/J_{f} and its Hilbert series

(1.1)  HP (M(f); t) = ∑_{k} dim M(f)_{k}t^{k}.

Assume in this note that the projective hypersurface V = V (f) : f = 0 in ℙ^{n} is reduced with (at most) isolated singularities at the points a_{1}, ..., a_{p}. Let Hess(f) be the Hessian matrix (f_{ij}) of the second order partial derivatives of f and h(f) be the Hessian of f, i.e. the determinant of this matrix Hess(f). This Hessian h(f) is a homogeneous polynomial in S of degree

1 Partially supported by Institut Universitaire de France and IAS Princeton.

T = (n + 1)(d − 2).

More generally, for each k satisfying 1 ≤ k ≤ n + 1 we denote by h_{k}(f) the ideal in S generated by all k × k-minors in the matrix Hess(f). In particular, the ideal h_{n+1}(f) = (h(f)) is a principal ideal. For each k as above, consider the graded k-th Hessian algebra of the polynomial f defined by

(1.2)  H_{k}(f) = S/(J_{f} + h_{k}(f)),

whose isomorphism class is clearly a GL(n + 1, ℂ)-invariant of f. Clearly we have a sequence of epimorphisms of graded ℂ-algebras

M(f) → H_{n+1}(f) → H_{n}(f) → ⋯ → H_{1}(f).

On the other hand, we introduce for any isolated hypersurface singularity (V, 0) at the origin of ℂ^{n}, given by an analytic germ g = 0, the k-th Hessian ideal h_{k}(g) to be the ideal generated by all the k-th minors in the Hessian (n × n)- matrix Hess(g) = (g_{ij}). By convention, we set h_{n+1}(g) = 0. It is easy to check that the isomorphism class of the local k-th Hessian algebra of the germ g defined by

(1.3)  H_{k}(g) = O_{n}/((g) + J_{g} + h_{k}(g)),

is a K-invariant of g, i.e. depends only on the isomorphism class of the germ (V, 0), see Lemma 2.1. Clearly we have now a sequence of epimorphisms of local Artinian ℂ-algebras

T (g) = H_{n+1}(g) → H_{n}(g) → ⋯ → H_{1}(g),

starting with the Tjurina algebra T (g) of g. Hence one can consider the Hessian algebras H_{k}(g) as a generalization of the Tjurina algebra T (g). For k = 1, ..., n + 1 we introduce the k-th Hessian number of g to be

(1.4)  χ_{k}(g) = dim H_{k}(g).

In view of the above remark, we can also write χ_{k}(g) = χ_{k}(V, 0) and use any normal form of g to compute these new invariants. Note that χ_{n+1}(V, 0) = τ(V, 0), the usual Tjurina number of (V, 0) and one has the inequalities

0 ≤ χ_{1}(V, 0) ≤ ... ≤ χ_{n}(V, 0) ≤ χ_{n+1}(V, 0) = τ(V, 0).

Our first main result is the following.

Theorem 1.1. Assume that V (f) is a hypersurface in ℙ^{n} with isolated singularities at the points a_{1}, ..., a_{p}. Then for k = 1, 2, ..., n + 1 and any m large enough one has

dim H_{k}(f)_{m} = ∑_{i=1,p} χ_{k}(V (f), a_{i}).

The following result estimates how large m should be in order to have such stabilization results. First we recall some definitions, see [12].

Definition 1.2. For a hypersurface V : f = 0 with isolated singularities we introduce three integers, as follows.

(i) the coincidence threshold

ct(V) = max{q : dim M(f)_{k} = dim M(f_{s})_{k} for all k ≤ q},

with f_{s} a homogeneous polynomial in S of degree d such that V_{s} : f_{s} = 0 is a smooth hypersurface in ℙ^{n}. (ii) the stability threshold st(V) = min{q : dim M(f)_{k} = τ(V) for all k ≥ q}, where τ(V) is the total Tjurina number of V, that is τ(V) = ∑_{i=1,p} τ(V (f), a_{i}). (iii) the minimal degree of a syzygy mdr(V) = min{q : H^{n}(K^{∗}(f))_{q+n} ≠ 0}, where K^{∗}(f) is the Koszul complex of f_{0}, ..., f_{n} with the natural grading.

It is known that one has

(1.5)  ct(V) = mdr(V) + d − 2,

and precise estimates on mdr(V) are given in [8], [9], [10], [13]. Our second main result is the following.

Theorem 1.3. Assume that the hypersurface V (f) in ℙ^{n} has only isolated singularities. Then one has the following stabilization property:

dim H_{n}(f)_{m} = τ(V (f)) − |Sing_{wh}(V (f))|

for any m large, where |Sing_{wh}(V (f))| is the number of weighted homogeneous singularities the hypersurface V (f) has. Moreover, for any integer k, 1 ≤ k ≤ n, the dimension of H_{k}(f)_{m} is constant for any

m ≥ T_{k} := max(T − ct(V) + k(d − 2), st(V)).

We also have the following result, expressing HP (H_{n+1}(f); t) in term of the series HP (M(f); t), which was already studied in [9], [10], [12].

Proposition 1.4. (i) Assume the hypersurface V (f) is smooth. Then

HP (H_{n+1}(f); t) = HP (M(f); t) − 1 = \frac{(1 − t^{d−1})^{n+1}}{(1 − t)^{n+1}} − 1.

(ii) Assume the hypersurface V (f) has isolated singularities (and it is not smooth). Then

HP (H_{n+1}(f); t) = HP (M(f); t).

A consequence of these results is the following (possibly the fastest) way to compute the total number of singularities of a projective hypersurface in cases when it is known a priori that these singularities are all weighted homogeneous.

Corollary 1.5. Assume that the hypersurface V (f) in ℙ^{n} has only isolated singularities (and it is not smooth). Then the total number of weighted homogeneous singularities of V (f) is given by

|Sing_{wh}(V (f))| = dim M(f)_{m} − dim H_{n}(f)_{m}

for any m ≥ T̃_{n} = (2n + 1)(d − 2).

In the second section we prove these results, give some Corollaries and a number of Examples. In the final section we offer a discussion on the constructible partitions induced on the space of homogeneous polynomials of a fixed degree d by the values of the Hilbert-Poincaré series of the Milnor algebra M(f) and of the Hessian algebras H_{k}(f). The only results here are of an experimental computational nature. The computations of various invariants given in this paper were made using two computer algebra systems, namely CoCoA [3] and Singular [5]. The corresponding codes are available on request. We would like to thank the referee for the careful reading of our manuscript and his useful remarks.

2. Hessian matrix and Hessian ideals

Lemma 2.1. The isomorphism class of the local k-th Hessian algebra

H_{k}(g) = O_{n}/((g) + J_{g} + h_{k}(g)),

of the isolated hypersurface germ (V, 0) : g = 0 is a K-invariant of g for any integer k = 1, 2, ..., n.

Proof. We have to show that if u ∈ O_{n} is a unit and ϕ : (ℂ^{n}, 0) → (ℂ^{n}, 0) an isomorphism germ, and if we set g′(y) = u(y) · g(ϕ(y)), then the algebra H_{k}(g′) is isomorphic to the algebra H_{k}(g) for any k. Note first that the quotients O_{V} = O_{n}/(g) and O_{V′} = O_{n}/(g′) are isomorphic ℂ-algebras, since the isomorphism ϕ^{∗} : O_{n} → O_{n} given by a(y) ↦ a(ϕ(y)) sends the ideal (g) injectively onto the ideal (g′). Computing the partial derivatives of g′ with respect to y_{1}, ..., y_{n}, we see that the isomorphism ϕ^{∗} sends also the ideal (g)+J_{g} into the ideal (g′)+J_{g′}. The inclusion ϕ^{∗}((g) + J_{g}) ⊂ (g′) + J_{g′} implies τ(g) ≥ τ(g′). Since the relation between g and g′ is symmetrical, we get in the same way τ(g′) ≥ τ(g), and hence in fact we have equalities everywhere. Now compute the second order partial derivatives of g′ and note that, modulo the ideal (g′) + J_{g′}, the Hessian matrix Hess(g′) is a product of 3 matrices A(y)^{t}Hess(g)(ϕ(y))A(y), where A(y) = dϕ(y) is the Jacobian matrix of ϕ and A(y)^{t} denotes its transpose. From this relation it follows that

ϕ^{∗}((g) + J_{g} + h_{k}(g)) ⊂ (g′) + J_{g′} + m_{k}(g′)

for any k. The equality is established exactly as above, using the symmetry between g and g′.

□

2.2. Proof of Theorem 1.1. For any k, it is clear that the ideal J_{f} + h_{k}(f) defines a 0-dimensional subscheme in ℙ^{n} whose support is contained in the singular locus V (f)_{sing} = {a_{1}, ..., a_{p}}. Hence one has for large m the following equality

(2.1)  dim H_{k}(f)_{m} = ∑_{i=1,p} dim H_{k}(f)_{a_{i}},

where H_{k}(f)_{a_{i}} denotes the analytic localization of the graded S-module H_{k}(f) at the point a_{i}, exactly as in the proof of Corollary 9 in [2].

To study such a localization, assume that the coordinates have been chosen such that the singular point a = a_{1} is located at (1 : 0 : ... : 0). Then the isolated hypersurface singularity (V (f), a) has a local equation

g(y_{1}, ..., y_{n}) = f(1, y_{1}, ..., y_{n}) = 0.

It follows that g_{j} = f_{j}(1, y_{1}, ..., y_{n}) for any j = 1, ..., n, where g_{j} denotes the partial derivative of g with respect to y_{j}. Using the Euler relation for f exactly as in the proof of Corollary 9 in [2], we also get

(2.2) d · g = f_{0}(1, y_{1}, ..., y_{n}) + y_{1}f_{1}(1, y_{1}, ..., y_{n}) + ... + y_{n}f_{n}(1, y_{1}, ..., y_{n}).

This implies that the localization of the Jacobian ideal J_{f} at the point a coincides with the ideal spanned by g, g_{1}, ..., g_{n}, i.e. the Tjurina ideal of the isolated hypersurface singularity (V (f), a). Let’s look now at the second order derivatives. One clearly has g_{ij} = f_{ij}(1, y_{1}, ..., y_{n}) for all 1 ≤ i, j ≤ n. If we take the partial derivative with respect to y_{j} for some j > 0 of the polynomials entering into the equality (2.2), we get

(2.3)  d · g_{j} = f_{0j}(1, y_{1}, ..., y_{n}) + y_{1}f_{1j}(1, y_{1}, ..., y_{n}) + ...+
+(f_{j}(1, y_{1}, ..., y_{n}) + y_{j}f_{jj}(1, y_{1}, ..., y_{n})) + ... + y_{n}f_{nj}(1, y_{1}, ..., y_{n}).

Similarly, starting with the Euler relation for f_{0}, taking the partial derivative with respect to x_{0} and then setting x_{0} = 1 and x_{j} = y_{j} for j > 0 we get

(2.4) (d − 1) · f_{0}(1, y_{1}, ..., y_{n}) =

= (f_{0}(1, y_{1}, ..., y_{n}) + f_{00}(1, y_{1}, ..., y_{n})) + ∑_{j>0} y_{j}f_{0j}(1, y_{1}, ..., y_{n}).

The formulas (2.3) and (2.4) give a proof of the following result.

Proposition 2.3. The localization of the ideal J_{f} +h_{k}(f) at the singular point a = (1 : 0 : ... : 0) coincides with the ideal in the corresponding local ring O_{n} spanned by the local equation g(y) = f(1, y_{1}, ..., y_{n}), its partial derivatives g_{j} with respect to y_{j} for j = 1, ..., n and all k ×k minors in the local n×n Hessian matrix Hess(g) = (g_{ij}).

Proof. Indeed, modulo the Tjurina ideal, the matrix Hess(f)(1, y_{1}, ..., y_{n}) can be transformed using the above formulas by obvious line and column operations into a matrix obtained from the local Hessian matrix Hess(g) by adding a first line and a first column formed by zeros only.

□

Combining this definition with Proposition 2.3 and the equality (2.1) clearly completes the proof of Theorem 1.1.

Example 2.4. Assume that (V, 0) is an A_{1}-singularity, hence a node. Then we can choose an equation of the form g = y_{1}^{2} + ... + y_{n}^{2} and it is clear that χ_{k}(V, 0) = 0 for k = 1, 2, ..., , n and χ_{n+1}(V, 0) = 1. In particular, if V (f) is a nodal hypersurface then H_{k}(f)_{m} = 0 for 1 ≤ k ≤ n and m large. Hence the corresponding Hilbert-Poincaré series

HP (H_{k}(f); t) = ∑_{m} dim H_{k}(f)_{m}t^{m}

is a polynomial for 1 ≤ k ≤ n.

Example 2.5. Assume that n = 2 and (V, 0) is an A_{q}-singularity. Then we can choose an equation of the form g = y_{1}^{2} + y_{2}^{q+1} and it is clear that χ_{1}(V, 0) = 0, χ_{2}(V, 0) = q − 1 and χ_{3}(V, 0) = q. In particular, if V (f) is a cuspidal curve then dim H_{k}(f)_{m} = (k − 1)κ for m large, where κ is the number of cusps A_{2}. To have a numerical example, let f = x^{2}y^{2} + y^{2}z^{2} + x^{2}z^{2} − 2xyz(x + y + z). Then f = 0 has κ = 3 cusps and no other singularities, see also [6], p.129 for more on this quartic curve. Numerical computations show that

HP (H_{1}(f); t) = 1 + 3t = 1 + 3t + 0(t^{2} + t^{3} + ...)
HP (H_{2}(f); t) = 1 + 3t + 6t^{2} + 7t^{3} + 3(t^{4} + t^{5} + ...)
HP (H_{3}(f); t) = 1 + 3t + 6t^{2} + 7t^{3} + 6(t^{4} + t^{5} + ...).

In particular

dim H_{k}(f)_{m} = 3(k − 1)

for k = 1, 2, 3 and m ≥ 4.

Example 2.6. Assume that n = 2 and (V, 0) is an D_{q}-singularity, q ≥ 4. Then we can choose an equation of the form g = y_{1}^{2}y_{2} + y_{2}^{q−1} and it is clear that χ_{1}(V, 0) = 1, χ_{2}(V, 0) = q − 1 and χ_{3}(V, 0) = q. Moreover, it clear that for a plane curve singularity (V, 0), one has χ_{1}(V, 0) = 1, if and only if (V, 0) has type D_{q} for some q ≥ 4.

Using Theorem 1.1 and Example 2.5 we get the following.

Example 2.7. The curve V (f) in ℙ^{2} has only A_{q} singularities for various q’s (i.e. V (f) has only corank 1 singularities) if and only if H_{1}(f)_{m} = 0 for m large. If this holds, then the total number of singularities of V (f) is given by

|Sing(V (f))| = dim H_{3}(f)_{m} − dim H_{2}(f)_{m}

for m large.

2.8. Proof of Theorem 1.3. Consider the local case of an isolated weighted homogeneous singularity (V, 0) given by g = 0. Then g ∈ J_{g} and in the local Gorenstein ring M(g) = O_{n}/J_{g}, the Hessian h(g) of g spans the socle which is one dimensional, see [15]. Clearly we have H_{n}(g) = M(g)/(h(g)), and hence χ_{n}(g) = τ(g) − 1 = χ_{n+1}(g) − 1. Next, if g is not weighted homogeneous, then the ideal (g) in M(g) is nonzero, and hence contains the minimal ideal (h(g)). It follows that in such a case χ_{n}(g) = τ(g). The result then follows for large m using Theorem 1.1 . Let I be the saturation of the Jacobian ideal J_{f}. Then Prop. 2. in [7] tells us that

(2.5)  dim S_{q}/I_{q} = τ(V (f))

for any q ≥ T − ct(D). In other words, for such q’s, the evaluation map

(2.6)  ev_{q} : S_{q}/I_{q} → ⊕O_{n}/((g) + J_{g})

is a bijection, where the sum is taken over all singularities (V, a) : g = 0 of the hypersurface V = V (f). In particular, for any fixed singularity a ∈ V_{sing} there is a polynomial h_{a} ∈ S_{q} such that the class of h_{a} in the summand T (g) = O_{n}/((g) + J_{g}) corresponding to a is 1 and its classes in all the other summands are 0. The k-minors in the matrix Hess(f) have degree k(d − 2) and the computation done in Proposition 2.3 shows that for any singular point a ∈ V_{sing} there are linear combinations m_{s} of such minors (arising from a change of coordinates on ℙ^{n}) such that the classes of m_{s} in the corresponding local ring T (g) generate the ideal h_{k}(g). Corollary 2 in [7] tells us that

(2.7)  I_{q} = J_{f,q}

for all q ≥ q_{1} = max(T − ct(V (f)), st(V (f))). Consider the bijection ev_{q} for q ≥ T_{k} := max(T − ct(V) + k(d − 2), st(V)) ≥ q_{1}. Since all the products h_{a}m_{s} ∈ h_{k}(f)_{q}, it follows that

ev_{q}((J_{f,q} + h_{k}(f)_{q})/J_{f,q}) = ⊕((g) + J_{g} + h_{k}(g)/((g) + J_{g}).

This gives the following bijection for q ≥ T_{k}

ev_{q} : H_{k}(f)_{q} → ⊕O_{n}/((g) + J_{g} + h_{k}(g))

which completes the proof of Theorem 1.3.

Remark 2.9. Note that T_{k} ≤ T̃_{k} = T + k(d − 2) = (n + k + 1)(d − 2) for any k. So one can use this larger bound when there is no information on ct(V) and st(V), i.e. we always have stabilization for dim H_{k}(f)_{m} when m ≥ T̃_{k}. When we have information on ct(V) but not on st(V), as it is often the case, one can use the better general bound

T̂_{k} = T + max(k(d − 1) − ct(V), 0).

Corollary 2.10. (i) The hypersurface V (f) in ℙ^{n} is nodal if and only if H_{n}(f)_{m} = 0 for m ≥ T̂_{n} = 3n(d − 2)/2. In particular, for n = 2, we have H_{2}(f)_{m} = 0 for m ≥ T = T̂_{2} = 3d − 6. (ii) Assume the hypersurface V (f) in ℙ^{n} is nodal and n ≥ 4. Then H_{n−1}(f)_{m} = 0 for m ≥ T̂_{n−1} = (3n − 2)(d − 2)/2.

Proof. For a nodal hypersurface, it is known that ct(V (f)) ≥ (n + 2)(d − 2)/2, see [9]. It follows that

T − ct(V (f)) + n(d − 2) ≤ T + (n − 2)(d − 2)/2 = 3n(d − 2)/2 ≥ T,

which proves the result (i). The second claim (ii) follows along the same lines. The condition n ≥ 4 is necessary to insure that T̂_{n−1} ≥ T.

□

Example 2.11. For the nodal curves f = xyz^{d−2} + x^{d} + y^{d} = 0, a direct computation shows that the stabilization H_{2}(f)_{m} = 0 occurs for m ≥ 3d − 7 for low values of d, so only one unit better than our prediction. For the Chebyshev curves considered in [11] or in [22], a direct computation shows that the stabilization H_{2}(f)_{m} = 0 occurs for m ≥ 3d − 8 for low values of d, so just 2 units better than our general prediction in Corollary 2.10.

Example 2.12. Let V : f = 0 be the Chebyshev 4-fold of degree d in ℙ^{5} considered in [10] or in [22]. A direct computation shows that the stabilization H_{4}(f)_{k} = 0 occurs for k ≥ 13m − 10 for low values of d = 2m + 1, and for k ≥ 13m − 17 for low values of d = 2m. The corresponding (integral) values given by T′_{4} in these cases are 13m−7 for low values of d = 2m+1 and 13m−13 for low values of d = 2m, so the difference between the theoretical bound and the real value of k is less than 4 for any degree d. Similarly, the stabilization H_{5}(f)_{k} = 0 occurs for k ≥ 15m − 10 for low values of d = 2m + 1, and for k ≥ 15m − 17 for low values of d = 2m. The corresponding (integral) values given by T′_{5} in these cases are 15m−8 for low values of d = 2m+1 and 15m−15 for low values of d = 2m, so the difference between the theoretical bound and the real value of k is less than 2 for any degree d.

Corollary 2.13. Assume the curve V (f) in ℙ^{2} has ν nodes and κ cusps as singularities. Then dim H_{2}(f)_{m} = κ and dim M(f)_{m} = dim H_{3}(f)_{m} = ν + 2κ for m ≥ T + d/6. In other words, the asymptotic behavior of dim H_{k}(f)_{m} for k = 2, 3 determines the number of nodes and cusps.

Proof. For a plane curve with nodes and cusps, it is known that ct(V (f)) ≥ 11d/6 − 4, see [13]. It follows that T_{1} ≤ T + d/6, which proves the result.

□

The stabilization value of H_{n−1}(f)_{m} is not easy to describe gemetrically beyond Theorem 1.1, as the following example shows.

Example 2.14. Let V : f = 0 be a plane curve having only simple singularities, hence singularities of type A_{k}, D_{k}, E_{6}, E_{7} and E_{8}. A direct computation shows that χ_{1}(g) = 2 for g of type E_{6} and E_{7}, and χ_{1}(g) = 3 for g of type E_{8}. Using now Example 2.6, it follows that, for m ≥ T̃_{1} = 4d − 8, one has

dim H_{1}(f)_{m} = ∑_{k} ♯D_{k} + 2(♯E_{6} + ♯E_{7}) + 3♯E_{8}.

Example 2.15. Consider the line arrangement in ℙ^{2} given by the following equation

f = (x^{2} − y^{2})(y^{2} − z^{2})(x^{2} − z^{2}).

Then the curve V (f) : f = 0 is a union of 6 lines and it has 3 nodes A_{1} and 4 triple points D_{4}. In this case ct(V) = 6 as shown by HP (H_{3}(f); t) = HP (M(f); t), and hence T_{2} = max(12 − 6 + 8, st(V)) = 14 and T_{1} = T_{2} − (d − 2) = 10. Numerical computations give the following.

HP (H_{1}(f); t) = 1 + 3t + 6t^{2} + 10t^{3} + 10t^{4} + 7t^{5} + 5t^{6} + 4(t^{7} + t^{8} + ...)

(hence again a stabilization 3 units before T_{1})

HP (H_{2}(f); t) = 1+3t+6t^{2}+10t^{3}+15t^{4}+18t^{5}+19t^{6}+19t^{7}+13t^{8}+12(t^{9}+t^{10}+...)

(hence again a stabilization 2 units before T_{2}) and

HP (H_{3}(f); t) = HP (M(f); t) = 1+3t+6t^{2}+10t^{3}+15t^{4}+18t^{5}+19(t^{6}+t^{7}+...),

as predicted by Proposition 1.4.

The following consequence of the proof of Theorem 1.3 might be useful sometimes.

Corollary 2.16. An isolated hypersurface singularity g = 0 is weighted homogeneous if and only if χ_{n}(g) = τ(g) − 1. Otherwise χ_{n}(g) = τ(g).

2.17. Proof of Proposition 1.4. In the case (i), it is enough to note that M(f) is an Artinian Gorenstein ring whose 1-dimensional socle is spanned by the Hessian polynomial h(f) which has degree T, see for instance [15].

In the case (ii), the class of the Hessian h(f) will now vanish in the local Tjurina ring T (g) attached to any singularity of V (f) by Proposition 2.3. Indeed, recall that the first row and first column are zero in the Hessian matrix Hess(f) (in suitable coordinates) modulo the Tjurina ideal (g) + J_{g}. Moreover, as in the proof of Theorem 1.3, the evaluation map

(2.8)  ev_{T} : S_{T} /J_{f,T} → ⊕O_{n}/((g) + J_{g})

is an isomorphism. It follows that h(f) ∈ J_{f} and hence HP (H_{n+1}(f); t) = HP (M(f); t) in this case.

The Hilbert-Poincaré series of the Hessian rings H_{k}(f) for 1 ≤ k ≤ n are quite mysterious, as the following example shows.

Example 2.18. Consider the Fermat curve V (f) : f = x^{6} + y^{6} + z^{6} = 0 and its ”deformation” V (f′) : f′ = f + 2xyz^{4} + 3xzy^{4} + 5yzx^{4} = 0. One obtains the following results by direct computation.

(2.9) HP (M(f); t) = HP (M(f′); t) =

= 1+3t+6t^{2} +10t^{3} +15t^{4} +18t^{5} +19t^{6} +18t^{7} +15t^{8} +10t^{9} +6t^{10} +3t^{11} +t^{12}.

In particular, V (f′) is smooth as well and HP (H_{3}(f); t) = HP (H_{3}(f′); t) can be computed using Proposition 1.4. For k = 2 we get the following results.

HP (H_{2}(f); t) = 1+3t+6t^{2} +10t^{3} +15t^{4} +18t^{5} +19t^{6} +18t^{7} +12t^{8} +7t^{9} +3t^{10}

and

HP (H_{2}(f′); t) = 1 + 3t + 6t^{2} + 10t^{3} + 15t^{4} + 18t^{5} + 19t^{6} + 18t^{7} + 9t^{8}.

Note that the 2-minors have degree 8, which explains the fact that the monomials containing t^{m} for m < 8 are unchanged. For f there are only 3 such minors, which explain the drop from 15 to 12 in the coefficient of t^{8}. For f′, there are 6 distinct minors (due to the symmetry of the Hessian matrix) and these 2-minors span a 6-dimensional vector space modulo J_{f′,8} in this case. Finally, for k = 1 we get

HP (H_{1}(f); t) = 1 + 3t + 6t^{2} + 10t^{3} + 12t^{4} + 12t^{5} + 10t^{6} + 6t^{7} + 3t^{8} + t^{9}

and

HP (H_{1}(f′); t) = 1 + 3t + 6t^{2} + 10t^{3} + 9t^{4} + 3t^{5}.

Note that the 1-minors have degree 4, which explains the fact that the monomials containing t^{m} for m < 3 are unchanged. For f there are only 3 such minors, which explain the drop from 15 to 12 in the coefficient of t^{4}. For f′, there are 6 distinct 1-minors (due to the symmetry of the Hessian matrix) and these 1-minors are linearly independent in this case as seen by looking at the coefficient of t^{4}. Moreover, if we look at the coefficient of t^{5} in HP (H_{1}(f′); t), it says that the only relations among the 18 polynomials x_{m}f′_{ij} with i ≤ j and the 3 partial derivatives f′_{0}, f′_{1}, f′_{3} are the obvious 3 relations given by the Euler’s formulas

(d − 1)f′_{q} = x_{0}f′_{q0} + x_{1}f′_{q1} + x_{2}f′_{q2},

for q = 0, 1, 2. This accounts for the drop 18 − 15 = 3 when we pass from M(f′) to H_{1}(f′). The symmetry of the polynomial HP (H_{1}(f); t) is easy to explain in general. Indeed, if we denote by F (d) the Fermat polynomial in n+1 variables of degree d, then it is clear that

H_{1}(F (d)) = M(F (d − 1)).

3. Constructible partitions for the space of homogeneous polynomials

Consider the projective space ℙ(S_{d}) of non-zero homogeneous polynomials of degree d and let G = P GL_{n}(ℂ) act on ℙ(S_{d}) in the usual way, i.e. by linear changes of coordinates. Then there is a Zariski open and dense, G-invariant subset U_{s}(n, d) in ℙ(S_{d}), parametrizing the smooth hypersurfaces of degree d in ℙ^{n}. There is a larger G-invariant Zariski open subset U_{iso}(n, d) such that f ∈ U_{iso}(n, d) if and only if the hypersurface V (f) : f = 0 in ℙ^{n} has only isolated singularities. Moreover, for each list of (distinct or not) simple isolated hypersurface singularities (Y_{1}, Y_{2}, ..., Y_{p}) we can consider the constructible subset S(Y_{1}, Y_{2}, ..., Y_{p}) ⊂ U_{iso}(n, d) defined by the condition that f ∈ S(Y_{1}, Y_{2}, ..., Y_{p}) if and only if the hypersurface V (f) has exactly p singularities of types Y_{1}, ..., Y_{p}. It is known that if this stratum is non-empty, then

codim S(Y_{1}, Y_{2}, ..., Y_{p}) ≤ ∑_{i=1,p} τ(Y_{i})

with equality if the degree d is large, e.g. if

(3.1)  d ≥ ∑_{i=1,p} s_{i} + p − 1,

where s_{i} is the K-determinacy order of Y_{i}, see Prop. 1.3.9 in [6], p.18. Moreover in this case the stratum S(Y_{1}, Y_{2}, ..., Y_{p}) is smooth and connected. For more precise and more general results, see Theorem 1 and Theorem 5 in [16]. When all of the singularities Y_{i} are nodes and n = 2, then the corresponding stratum S_{0}(Y_{1}, Y_{2}, ..., Y_{p}) consisting of all reduced irreducible curves with p nodes is connected. This is the famous Severi problem solved by J. Harris [18]. However, in general, the strata S(Y_{1}, Y_{2}, ..., Y_{p}) are not connected even in this case, see Example 3.1 below. For more general and precise results in this direction, as well as for an extensive bibliography on the subject, we refer to the excellent survey [16]. Since dim M(f)_{k} = τ(V (f)), for k > T = (n + 1)(d − 2), see [2], it follows that there is a finite constructible Jacobian partition S_{Y} (M(f)) of each stratum S(Y) = S(Y_{1}, Y_{2}, ..., Y_{p}), such that f, g ∈ S(Y) are in the same stratum of S_{Y} if and only if HP (M(f); t) = HP (M(g); t). It is known that f_{s} ∈ U_{s}(n, d) if and only if

(3.2)  HP (M(f_{s}); t) = \frac{(1 − t^{d−1})^{n+1}}{(1 − t)^{n+1}} ,

and hence the Jacobian partition on U_{s}(n, d) consists of just one stratum. However, for a nodal hypersurface V (f) with 3 nodes, the value of HP (M(f); t) depends on whether the nodes are collinear or not, see Example 4.3 (iii) in [12]. It follows that the Jacobian stratification on the stratum S(3A_{1}) is not trivial in general.

Example 3.1. Consider the case n = 2, d = 4 and the stratum S(3A_{1}). It is known that this stratum S(A_{3}) has two irreducible components, X = S_{0}(3A_{1}) containing the irreducible quartics with 3 nodes (necessarily non-collinear) and Y, containing the reduced quartics with 3 collinear nodes obtained from a smooth cubic and a trisecant line, see Remark 4.7.21 in [21]. Then one has codim X = codim Y = 3, as follows from Theorem 4.7.18 in [21]. Example 4.3 (iii) in [12] tells us that there are two possibilities for H(M(f); t) on the stratum S(3A_{1}), namely the minimal value (coefficient by coefficient) is

HP (M(f); t) = 1 + 3t + 6t^{2} + 7t^{3} + 6t^{4} + 3t^{5} + 3t^{6} + ...

and this is attained exactly on X and the other possible value is

HP (M(g); t) = 1 + 3t + 6t^{2} + 7t^{3} + 6t^{4} + 4t^{5} + 3t^{6} + ...

and which is attained precisely on Y. In other words, in this case the Jacobian partition separates the irreducible components of the stratum S(3A_{1}). Consider now the case n = 2, but d ≥ 8. Since the K-determinacy order of a node is s_{i} = 2, it follows from (3.1) that the stratum S(3A_{1}) is smooth and connected in this case. The minimal value of HP (M(f); t), which is

(3.3)  \frac{(1 − t^{d−1})^{3}}{(1 − t)^{3}} + 2t^{T} + 3t^{T +1}(1 + t + t^{2} + ...),

by Example 4.3 (iii) in [12], is attained on a Zariski open dense subset S′ of the stratum S(3A_{1}), while the larger value

(3.4)  \frac{(1 − t^{d−1})^{3}}{(1 − t)^{3}} + t^{T −1} + 2t^{T} + 3t^{T +1}(1 + t + t^{2} + ...),

is attained on the closed subset S′′ = S(3A_{1}) \ S′ in S(3A_{1}).

Example 3.2. Consider the case n = 2, d = 6 and the stratum S(6A_{2}). It is known by the classical work of Zariski that this stratum has two irreducible components, X containing the irreducible sextics with 6 cusps on a conic and Y, containing the irreducible sextics with 6 cusps not on a conic. As an example of a sextic in X one can take

V (f) : f = (x^{2} + y^{2})^{3} + (y^{3} + z^{3})^{2} = 0

and then a direct computation yields

(3.5) HP (M(f); t) =

= 1 + 3t + 6t^{2} + 10t^{3} + 15t^{4} + 18t^{5} + 19t^{6} + 18t^{7} + 16t^{8} + 13t^{9} + 12(t^{10} + · · · .

As an example of a sextic in Y one can take

V (g) : g = 27(x+y)^{3}(x+y−z)^{2}(x+y+2z)−27(x+y)^{2}(x+y−z)^{2}((x−y)^{2}−z^{2})+

+9((x + y)^{2} − z^{2})((x − y)^{2} − z^{2})^{2} − ((x − y)^{2} − z^{2})^{3} = 0,

see [20], formula (5.7), and again a direct computation yields

HP (M(g); t) = 1+3t+6t^{2} +10t^{3} +15t^{4} +18t^{5} +19t^{6} +18t^{7} +15t^{8} +12(t^{9} +....

We do not know whether the Hilbert-Poincaré series is constant on X and/or Y, but it seems that again it can separate distinct irreducible components in a stratum.

We have seen above, especially in Example 2.18, that though one can also define the Hessian partition by looking at the strata inside a given S(Y_{1}, ..., Y_{p}) where the corresponding Hilbert-Poincaré series HP (H_{k}(f); t) are constant for k = 1, 2, ..., n + 1, the behavior of these partitions is rather mysterious. Indeed, Example 2.18 shows that already the Hessian partition on the stratum U_{s}(n, d) can be quite subtle. To investigate the possibilities for the Hilbert series for HP (M(f); t) as well as for HP (H_{k}(f); t) for various k’s, in the neighborhood of a given point g ∈ ℙ(S_{d}), one can proceed as follows. It is known that the tangent space to the G-orbit of g (where all the invariants are clearly preserved) is given by (J_{g})_{d}. Let h_{1}, ..., h_{q} be a (monomial) basis of M(g)_{d} and consider the family of polynomials

(3.6)  f = g + a_{1}h_{1} + ... + a_{q}h_{q},

with a_{j} ∈ ℂ, which is exactly a transversal to the G-orbit of g. If we choose the a_{j} small enough, the values obtained for HP (M(f); t) as well as for HP (H_{k}(f); t) for various k’s, will cover all the possibilities that occur in a neighborhood of g (in the strong topology). If we allow any values for a_{j}, they will give sometimes interesting values, even though not related to the fixed polynomial g (but covering this time a Zariski open neighborhood of g).

Remark 3.3. Consider the case of smooth plane curves of increasing degree d.

(i) When d = 2, then U_{s}(2, 2) is a single G-orbit, hence all the invariants HP (H_{k}(f)) are constant on U_{s}(2, 2), i.e. the Hessian partition is trivial. This fact clearly holds for any n ≥ 2 as well. (ii) When d = 3, it is known that any polynomial in U_{s}(2, 3) is G-equivalent to a Hesse normal form

f = x^{3} + y^{3} + z^{3} + 3axyz,

with a^{3} ≠ −1. This comes essentially from the fact that if g = x^{3} + y^{3} + z^{3} is the corresponding Fermat polynomial, then xyz is a basis for the 1-dimensional vector space M(g)_{3}. (iii) When d = 4, if we take g = x^{4} + y^{4} + z^{4}, then a basis for M(g)_{4} is given by the following 6 monomials

x^{2}y^{2}, x^{2}z^{2}, y^{2}z^{2}, x^{2}yz, xy^{2}z, xyz^{2}.

But doing computation with 6 parameter families seems too complicated for the moment. That is why in the following examples we take h_{1}, ..., h_{q} just a family of independent elements of M(g)_{d}, i.e. we explore what happens only in a fixed direction in the normal space to the orbit of g.

Example 3.4. Consider the case n = 2, d = 6, g = x^{6} + y^{6} + z^{6} and the 3-parameter family

E : f = g + ax^{4}yz + bxy^{4}z + cxyz^{4},

which is a partial transversal as explained above. We will describe all the possibilities for the series HP (H_{1}(f); t) when f has this form and f ∈ U_{s}(2, 6). By direct computation one gets the following Hessian strata in E, where for each strata we list the corresponding series HP (H_{1}(f); t) and the conditions on a, b, c to get such a series, up to the action of the permutation group on x, y, z.

S_{1} : 1 + 3t + 6t^{2} + 10t^{3} + 9t^{4} + 3t^{5} with a, b, c generic .

This is the largest stratum, a Zariski open dense subset in E = ℂ^{3}.

S_{2} : 1 + 3t + 6t^{2} + 10t^{3} + 9t^{4} + 4t^{5} with a = b ≠ 0 and c generic .

S_{3} : 1 + 3t + 6t^{2} + 10t^{3} + 9t^{4} + 6t^{5} with a = b = c generic .

S_{4} : 1 + 3t + 6t^{2} + 10t^{3} + 9t^{4} + 4t^{5} + t^{6} with a = b ≠ 0 and c = 0.

S_{5} : 1 + 3t + 6t^{2} + 10t^{3} + 9t^{4} + 6t^{5} + 2t^{6} with a, b, c a collection of finite nonzero values, for instance a = b = c = 2.

S_{6} : 1 + 3t + 6t^{2} + 10t^{3} + 9t^{4} + 7t^{5} + 3t^{6} + t^{7} with a = b = 0 and c generic .

S_{7} : 1 + 3t + 6t^{2} + 10t^{3} + 12t^{4} + 12t^{5} + 10t^{6} + 6t^{7} + 3t^{8} + t^{9} with a = b = c = 0.

In the next picture (a Hasse diagram as in the theory of hyperplane arrangements) we draw a line between S_{i} and S_{j} with S_{i} above S_{j} if the inequality HP (H_{1}(f^{i}); t) ≥ HP (H_{1}(f^{j}); t) (coefficient by coefficient) holds for some f^{i} ∈ S_{i} and f^{j} ∈ S_{j} and we call the resulting poset a Hessian poset. This inequality is a necessary condition such that the closure of the stratum S_{j} intersects the stratum S_{i}, by obvious semicontinuity properties. This condition is not sufficient, since for instance the stratum S_{5} consists of finitely many points, hence its closure coincides with S_{5}, and has empty intersection with the strata S_{6} and S_{7}. However, all the other strata S_{j} contain g in their closure. Note also that if, instead of looking at strata S_{j} inside E ∩ U_{s}(2, 6), we consider the corresponding strata S′_{j} in U_{s}(2, 6) defined by the same series HP (H_{1}(f); t), the above claim that S_{5} is finite it is no longer valid for S′_{5}, and hence it is possible that the closure of S′_{5} in U_{s}(2, 6) contains g. To decide such a question by this approach would involve computation with a family having dim M(g)_{6} = 19 parameters. On the positive side, the above computation shows that the strata S′_{1}, S′_{2}, S′_{3}, S′_{4} and S′_{6} contain g in their closure.

[figure: S_{1}; S_{2}; S_{3}; S_{4}; S_{5}; S_{6}; S_{7}]

Figure 1. The Hasse diagram of a Hessian poset in the case n = 2, d = 6

Here is an example for the smooth curves of degree 6, which can be regarded as a continuation of Example 2.18 and where we consider both series HP (H_{1}(f); t) and HP (H_{2}(f); t).

Example 3.5. Consider the case n = 2, d = 6. Let g = f′′ = x^{6} + y^{6} + z^{6} + 3x^{2}y^{2}z^{2}. Then the curve V (f′′) : f′′ = 0 is smooth and we have

HP (H_{2}(f′′); t) = 1 + 3t + 6t^{2} + 10t^{3} + 15t^{4} + 18t^{5} + 19t^{6} + 18t^{7} + 9t^{8}, hence the same series as for the polynomial f′ in Example 2.18 and

HP (H_{1}(f′′); t) = 1 + 3t + 6t^{2} + 10t^{3} + 9t^{4} + 6t^{5}, hence a strictly larger series than for the polynomial f′ since the coefficients of t^{5} satisfy 3 < 6. Consider now the 1-parameter deformation f′′_{a} = f′′ + axyz^{4} of f′′. For small values of a we get HP (H_{2}(f′′_{a}); t) = 1 + 3t + 6t^{2} + 10t^{3} + 15t^{4} + 18t^{5} + 19t^{6} + 18t^{7} + 9t^{8}, hence the same series as for the polynomials f′ and f′′. Moreover

HP (H_{1}(f′′_{a}); t) = 1+3t+6t^{2}+10t^{3}+9t^{4}+4t^{5}, hence a series between that of the polynomial f′ and that of polynomial f′′, since the corresponding coefficients of t^{5} satisfy 3 < 4 < 6. However, for the value a = 3 we get different results, namely

HP (H_{2}(f′′_{3}); t) = 1 + 3t + 6t^{2} + 10t^{3} + 15t^{4} + 18t^{5} + 19t^{6} + 18t^{7} + 9t^{8} + 2t^{9}, and

HP (H_{1}(f′′_{3}); t) = 1 + 3t + 6t^{2} + 10t^{3} + 9t^{4} + 4t^{5}, hence this polynomial has both series distinct from the minimal values given by the polynomial f′ in Example 2.18. Note that the series HP (H_{2}(f′′_{3}); t) is strictly bigger than the series HP (H_{2}(f′′); t), while the series HP (H_{1}(f′′_{3}); t) is stricly smaller than HP (H_{1}(f′′); t). The conclusion is that the strata of the Hessian partition to which belong f′′_{3} and respectively f′′ are not adjacent, i.e. their closures are disjoint.

Remark 3.6. It is not at all clear what the series HP (H_{k}(f); t)) tells us about the geometry of the smooth hypersurface V (f) : f = 0.

(i) In the curve case, when n = 2, the equations f = h(f) = 0 define the inflection points of the curve V (f), and by Bezout Theorem, their number is at most d(3d − 6), with equality for a generic curve, see for instance [1]. This number is lower for some curves, e.g. for the Fermat curve x^{d} + y^{d} + z^{d} = 0 we have only 3d inflection points (counted without multiplicities). However, the series HP (H_{k}(f); t)) do not seem to be related to such differences. Indeed, the two smooth curves V (f′) and V (f′′_{3}) of degree 6 in Example 3.5 have each 72 inflection points (by direct computation), but the corresponding series are different.

(ii) The subvariety of V (f) ⊂ ℙ^{n} given by the vanishing of the ideal h_{n+2−k}(f) corresponds to the closure of the Thom-Boardman singularity set S_{k}(ϕ_{f}) of the dual mapping ϕ_{f} : V (f) → ℙ^{n} associated to f. For a generic f, the stratum S_{k}(ϕ_{f}) is smooth, of codimension k(k + 1)/2 in V (f) (in particular empty if if n − 1 < k(k + 1)/2) and connected if n − 1 > k(k + 1)/2, see [1].

(iii) When d = 3, there is a graded Artinian Gorenstein algebra R(f) with socle degree 3 associated to f via the so-called Macaulay inverse systems, see [4], [17], [19]. By results in classical algebraic geometry in [14], the variety given by the vanishing of the ideal h_{k}(f) corresponds to the linear forms ℓ ∈ R(f)_{1} such that the multiplication by ℓ gives a linear map R(f)_{1} → R(f)_{2} of rank < k, see Lemma 6.1 in [4]. In this way relations to the Lefschetz properties of Artinian graded rings are established. For such properties, and a definition of generalized Hessians playing a key role in this theory, see [17], in particular Definition 3.75.

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Alexandru Dimca Univ. Nice Sophia Antipolis Lab. J.-A. Dieudonné CNRS UMR 7351 06100 Nice, France. dimca@unice.fr Gabriel Sticlaru Faculty of Mathematics and Informatics Ovidius University 900527 Constanta, Romania gabrielsticlaru@yahoo.com 706

Documenta Mathematica 20 (2015)