Some criteria to check if a projective hypersurfaces is smooth or singular

In this paper we present some properties for projective hypersurfaces, smooth and singular, to be criteria for identification. To make the decision with these criteria, we have included procedures written in Singular language.


Introduction
Let S = C[x 0 , ..., x n ] be the graded ring of polynomials in x 0 , , ..., x n with complex coefficients and denote by S d the vector space of homogeneous polynomials in S of degree d. For any polynomial f ∈ S d we define the Jacobian ideal J f ⊂ S as the ideal spanned by the partial derivatives f 0 , ..., f n of f with respect to x 0 , ..., x n .
The Hilbert-Poincaré series of a graded S-module M of finite type is defined by and it is known, to be a rational function of the form For a proof see, for instance, [3], p. 109. As soon as the hypersurface V (f ) acquires some singularities, the series HP (M(f )) is an infinite sum.
Beyond the smooth case, there is just one general situation in which we know explicit formulas for HP (M(f )). For polynomials f such that the saturated Jacobian ideal J f of the Jacobian ideal J f is a complete intersection ideal of multidegree (d 1 , ..., d n ), then it is known that see Proposition 4, in [5]. For some curves or surfaces, we can find effective singularities and classify. It is generally difficult to find all the singularities. In this paper we will check if there is at least one singular point and possibly how many they are. It is a very hard work to obtain manually these informations, even if we consider curves and surfaces with low degree. In the last part, we present procedures in Singular languages.

Smooth hypersurfaces
Smooth hypersurfaces have the same Hilbert-Poincaré series associated to Fermat type . To the best of our knowledge, the only results are for n = 2 : In this section we show explicit formulas for the coefficients a k . To compute these coefficients, we need additional informations.
Let be the sequences 111 . . . 111 with k bits and k − 1 spaces. If we keep only n spaces (and all other will be deleted), we have n + 1 groups of successive bits. There are obviously k−1 n ways of doing this.

Corollary 2.2. The number of solutions in nonnegative integers
Hence M k is the difference of B and the union of B 0 , B 1 , . . . , B n , based on inclusion-exclusion principle, we obtain: Formulas for some cases We need to compute the a k coefficients only for 0

Some criteria and examples
In this section we present some criteria to check if the projective hypersurfaces is singular or smooth.
3.1. Critical points. For some curves or surfaces, we can find effective singularities and classify. For low degree and simple polynomials, we can find the critical points by solving the system of equations: Because f is homogenous of degree r, by Euler's formula we have: Obviously, if all partial derivatives of polynomial f vanish at p, then the polynomial f vanishes at p too. Therefore, for finding singular points it is enough to get critical points and classify.
Here is the minimal graded free resolution of Milnor algebra M: To get the formulas for the Hilbert-Poincaré series, we start with the resolution (3.2) and get HP (M)(t) = HP Then we use the well-known formulas HP (N 1 ⊕N 2 )(t) = HP (N 1 )(t)+HP (N 2 )(t), HP (N p (−q))(t) = pt q HP (N)(t), HP (S)(t) = 1 (1−t) 3 and we obtain:

Remark
It is sufficient to find just one coefficient which is different in the two series to infer that V (f ) is singular.
Another criteria is based on degree T = (n + 1)(d − 2) of the polynomial for the smooth case. If dim(M T +1 ) > 0 the hypersurfaces is singular.

Complete intersection.
We present projective hypersurfaces V (f ) for which the saturated Jacobian ideal J f is a complete intersection of multidegree (d 1 , ..., d n ), so these V (f ) are singular and their Hilbert-Poincaré series HP (M(f ))(t) are infinite. We display the type of complete intersection (d 1 , ..., d n ), the generators for the saturated Jacobian ideal J f and the Hilbert-Poincaré series.

Number of singularities.
In this subsection we assume that V (f ) has at most isolated singularities.
We consider the following ideals: J = J f the jacobian ideal and I the radical of J with G 1 = std(J) and G 2 = std(I) their respective standard Groebner basis, mult 1 the degree of G 1 and mult 2 the degree of G 2 .
Proof. Indeed, when V (f ) has isolated singularities, then one has mult 1 = τ (V (f )), the total Tjurina number of V (f ), which is the sum of the Tjurina numbers τ (V (f ), p) over all singular points p of V (f ). This follows from the definition of the multiplicity of a homogeneous ideal and the equality dim M(f ) k = τ (V (f )), for k large, see [1]. Now, at each singular point we have the inequality τ (V (f ), p) ≤ µ(V (f ), p), and by taking the sum we get the inequality τ (V (f )) ≤ µ(V (f )) between the total Tjurina and the total Milnor numbers. Moreover, it is known that µ(V (f )) ≤ (d − 1) n , see for instance Proposition (3.25), page 90 as well as the discussion on page 161 in [4]. It follows that in fact, for singular hypersurfaces we have the stronger inequality For the smooth case, the Hilbert-Poincaré series is polynomial, Corollary 3.6. If the hypersurfaces is singular and mult 1 = mult 2 them all the singularities are nodes.
If there are two type of singularities on V (f ), say A p and A q it is possible to find the number x, y of each type of singularities, by solving the system: For some case, there are many configuration, for example, if mult 1 = 5 and mult 2 = 3 we have two configurations A 1 + A 2 and 2A 2 + A 3 .
In the following table we present some examples for projective surfaces f = 0 of degree three. polynomial f mult 1 mult 2 Type wxz + y 3 6 3 3A 2 w(xy + xz + yz) + xyz Computation of genus. The genus of a smooth irreducible curve defined by a polynomial of degree d is given by the formula: The genus of a singular irreducible projective curve C : f = 0 (i.e. the genus of its smooth model) is given by g(C) = g s − δ, where δ is the sum of all local delta-invariants δ p of the singularities p ∈ C. If C is smooth, then δ is 0.
The Hermitian curve x d + y d = z d and Fermat curve Remark Every irreducible projective curves with genus= (d−1)(d−2) 2 − 1 is either nodal of type A 1 or has exactly one cusp A 2 .
Several computer algebra packages are able to compute the genus of a plane curve In the following table we present some computations with procedure genus() from Singular library normal.lib.

Programs in Singular language
For mathematical computations, we can use any CAS (Computer Algebra System) software like Mathematica, Matlab or Maple, but for Algebraic Geometry, the best are perhaps Singular, Macaulay2 or CoCoA.
Singular is a package developed at the University of Kaiserslautern, see [2] and [8].