Fibers of rational maps and Rees algebras of their base ideals

We consider a rational map $\phi: \mathbb{P}_k^{m} \dashrightarrow \mathbb{P}_k^n$ that is a parameterization of an $m$-dimensional variety. Our main goal is to study the $(m-1)$-dimensional fibers of $\phi$ in relation to the $m$-th local cohomology modules of the Rees algebra of its base ideal.


Introduction
Let k be a field and φ : P m k P n k be a rational map. Such a map φ is defined by homogeneous polynomials f 0 , . . . , f n , of the same degree d, in a standard graded polynomial ring R = k[X 0 , . . . , X m ], such that gcd(f 0 , . . . , f n ) = 1. The ideal I of R generated by these polynomials is called the base ideal of φ. The scheme B := Proj(R/I) ⊂ P m k is called the base locus of φ. Let B = k[T 0 , . . . , T n ] be the homogeneous coordinate ring of P n k . The map φ corresponds to the k-algebra homomorphism ϕ : B −→ R, which sends each T i to f i . Then, the kernel of this homomorphism defines the closed image S of φ. In other words, after degree renormalization, k[f 0 , . . . , f n ] ≃ B/Ker(ϕ) is the homogeneous coordinate ring of S . The minimal set of generators of Ker(ϕ) is called its implicit equations and the implicitization problem is to find these implicit equations.
The implicitization problem has been of increasing interest to commutative algebraists and algebraic geometers due to its applications in Computer Aided Geometric Design as explained by Cox [1].
We blow up the base locus of φ and obtain the following commutative diagram The variety Γ is the blow-up of P m k along B, and it is also the Zariski closure of the graph of φ in P m k × P n k . Moreover, Γ is the geometric version of the Rees algebra R I of I, i.e., Proj(R I ) = Γ. As R I is the graded domain defining Γ, the projection π 2 (Γ) = S is defined by the graded domain R I ∩ k[T 0 , . . . , T n ], and we can thus obtain the implicit equations of S from the defining equations of R I . Besides the computation of implicit representations of parameterizations, in geometric modeling it is of vital importance to have a detailed knowledge of the geometry of the object and of the parametric representation one is working with. The question of how many times is the same point being painted (i.e., corresponds to distinct values of parameter) depends not only on the variety itself but also on the parameterization. It is of interest for applications to determine the singularities of the parameterizations. The main goal of this paper is to study the fibers of parameterizations in relation to the Rees and symmetric algebras of their base ideals. More precisely, we set π := π 2|Γ : Γ −→ P n k .
For every closed point y ∈ P n k , we will denote its residue field by k(y). If k is assumed to be algebraically closed, then k(y) ≃ k. The fiber of π at y ∈ P n k is the subscheme Suppose that m ≥ 2 and φ is generically finite onto its image. Then, the set consists of only a finite number of points in P n k . For each y ∈ Y m−1 , the fiber of π at y is an (m − 1)-dimensional subscheme of P m k and thus the unmixed component of maximal dimension is defined by a homogeneous polynomial h y ∈ R. One of the interesting problems is to establish an upper bound for y∈Y m−1 deg(h y ) in terms of d. This problem was studied in [2,3].
The paper is organized as follows. In Section 2, we study the structure of Y m−1 . Some results in this section were proved in [2]. The main result of this section is Theorem 2.5 that gives an upper bound for y∈Y m−1 deg(h y ) by the initial degree of certain symbolic powers of its base ideal. This is a generalization of [3,Proposition 1] where the first author only proved this result for parameterizations of surfaces φ : P 2 k P 3 k under the assumption that the base locus B is locally a complete intersection. More precisely, we have the following.
Theorem If there exists an integer s such that ν = indeg((I s ) sat ) < sd, then In particular, if indeg(I sat ) < d, then y∈Y m−1 deg(h y ) < d.
In Section 3, we study the part of graded m in X i of the m-th local cohomology modules of the Rees algebra with respect to the homogeneous maximal ideal m = (X 0 , . . . , X m ) The main result of this section is the following.
Theorem (Theorem 3.2) We have that N is a finitely generated B-module satisfying In the last section, we treat the case of parameterization φ : P 2 k P 3 k of surfaces. We establish a bound for the Castelnuovo-Mumford regularity and the degree of the B-module 2 Fibers of rational maps φ : P m k P n k Let n ≥ m ≥ 2 be integers and R = k[X 0 , . . . , X m ] be the standard graded polynomial ring over an algebraically closed field k. Denote the homogeneous maximal ideal of R by m = (X 0 , . . . , X m ). Suppose we are given an integer d ≥ 1 and n + 1 homogeneous polynomials f 0 , . . . , f n ∈ R d , not all zero. We may further assume that gcd(f 0 , . . . , f n ) = 1, replacing the f ′ i s by their quotient by the greatest common divisor of f 0 , . . . , f n if needed; hence, the ideal I of R generated by these polynomials is of codimension at least two. Set B := Proj(R/I) ⊆ P m k := Proj(R) and consider the rational map whose closed image is the subvariety S in P n k . In this paper, we always assume that φ is generically finite onto its image, or equivalently that the closed image S is of dimension m. In this case, we say that φ is a parameterization of the m-dimensional variety S .
Let Γ 0 ⊂ P m k × P n k be the graph of φ : P m k \ B −→ P n k and Γ be the Zariski closure of Γ 0 . We have the following diagram where the maps π 1 and π 2 are the canonical projections. One has where the bar denotes the Zariski closure. Furthermore, Γ is the irreducible subscheme of P m k × P n k defined by the Rees algebra Denote the homogeneous coordinate ring of P n k by B := k[T 0 , . . . , T n ]. Set with the grading deg(X i ) = (1, 0) and deg(T j ) = (0, 1) for all i = 0, . . . , m and j = 0, . . . , n. The natural bi-graded morphism of k-algebras is onto and corresponds to the embedding Γ ⊂ P m k × P n k . Let P be the kernel of α. Then, it is a bi-homogeneous ideal of S, and the part of degree one of P in T i , denoted by P 1 = P ( * ,1) , is the module of syzygies of the I a 0 T 0 + · · · + a n T n ∈ P 1 ⇐⇒ a 0 f 0 + · · · + a n f n = 0. Set S I := Sym R (I) for the symmetric algebra of I. The natural bi-graded epimorphisms Let K be the kernel of δ, one has the following exact sequence Notice that the module K is supported in B because I is locally trivial outside B.
As the construction of symmetric and Rees algebras commutes with localization, and both algebras are the quotient of a polynomial extension of the base ring by the Koszul syzygies on a minimal set of generators in the case of a complete intersection ideal, it follows that Γ and V coincide on (P m k \ X) × P n k , where X is the (possibly empty) set of points where B is not locally a complete intersection. Now we set π := π 2|Γ : Γ −→ P n k . For every closed point y ∈ P n k , we will denote its residue field by k(y), that is, k(y) = B p /pB p , where p is the defining prime ideal of y. As k is algebraically closed, k(y) ≃ k. The fiber of π at y ∈ P n k is the subscheme Our goal is to study the structure of Y ℓ . Firstly, we have the following.
Furthermore, this inequality is strict for any ℓ > 0. As a consequence, π has no mdimensional fibers and only has a finite number of (m − 1)-dimensional fibers.
The fibers of π are defined by the specialization of the Rees algebra. However, Rees algebras are difficult to study. Fortunately, the symmetric algebra of I is easier to understand than R I , and the fibers of π are closely related to the fibers of π ′ := π 2|V : V −→ P n k .
Recall that for any closed point y ∈ P n k , the fiber of π ′ at y is the subscheme The next result gives a relation between fibers of π and π ′ -recall that X is the (possible empty) set of points where B is not locally a complete intersection.
The fibers of π and π ′ agree outside X, hence they have the same (m − 1)-dimensional fibers.
The next result is a generalization of [4, Lemma 10] that gives the structure of the unmixed part of a (m − 1)-dimensional fiber of π. Note that our result does not need the assumption that B is locally a complete intersection as in [4], thanks to Lemma 2.2. Recall that the saturation of an ideal J of R is defined by J sat := J : R (m) ∞ .

Remark 2.4
The above lemma shows that the (m − 1)-dimensional fibers of π can only occur when B = ∅ as B ⊃ V (f i , h y ). It also shows that if there is a (m − 1)-dimensional fiber with unmixed part given by h y . As a consequence, deg(h y ) < d for any y ∈ Y m−1 .
By Lemma 2.1, π only has a finite number of (m−1)-dimensional fibers. The following give an upper bound for this number in terms of the initial degree of certain symbolic powers of its base ideal. Recall that the initial degree of a graded R-module M is defined by indeg(M) := inf{n ∈ Z | M n = 0} with convention that sup ∅ = +∞.
Theorem 2.5 If there exists an integer s ≥ 1 such that ν = indeg((I s ) sat ) < sd, then In particular, if indeg(I sat ) < d, then Proof. As Y m−1 is finite, by Lemma 2.3, there exists a homogeneous polynomial f ∈ I of degree d such that, for any y ∈ Y m−1 , I = (f ) + h y (g 1y , . . . , g ny ) and I sat ⊂ (f, h y ) for some g 1y , . . . , g ny ∈ R. Since (f, h y ) is a complete intersection ideal, it follows from [5, Appendix 6, Lemma 5] that (f, h y ) s is unmixed, hence saturated for every integer s ≥ 1. Therefore, for all y ∈ Y m−1 , . . , h s y ). Now, let 0 = F ∈ (I s ) sat such that deg(F ) = ν < sd, then h y is a divisor of F . Moreover, if y = y ′ in Y m−1 , then gcd(h y , h y ′ ) = 1. We deduce that

Remark 2.6
In the case where φ : P 2 k P 3 k is a parameterization of surfaces. In [3], the first author showed that if B is locally a complete intersection of dimension zero, then Example 2.7 Consider the parameterization φ : P 2 k P 3 k of surface given by Using Macaulay2 [6], it is easy to see that I = I sat and indeg((I 2 ) sat ) = 8 < 2.5 = 10. Furthermore, I admits a free resolution where matrix M is given by Thus, we obtain Y 1 = {p 1 , p 2 , p 3 , p 4 , p 5 , p 6 , p 7 , p 8 } with

Local cohomology of Rees algebras of the base ideal of parameterizations
Let φ : P m k P n k be a parameterization of m-dimensional variety. Let R = k[X 0 , . . . , X m ] and B = k[T 0 , . . . , T n ] be the homogeneous coordinate ring of P m k and P n k , respectively. For every closed point y ∈ P n k , the fiber of π at y is the subscheme π −1 (y) = Proj(R I ⊗ B k(y)) ⊂ P m k(y) ≃ P m k and we are interested in studying the set We now consider the B-module where m = (X 0 , . . . , X m ) is the homogeneous maximal ideal of R. By [7, Theorem 2.1], M µ is a finitely generated B-module for all µ ∈ Z. The following result gives a relation between the support of M µ and Y m−1 . For each y ∈ P n k = Proj(B), we can see y as a homogeneous prime ideal of B.

Proposition 3.1 One has
Proof. As k is algebraically closed, we have π −1 (y) = Proj(R I ⊗ B k(y)) ⊂ P m k(y) ≃ P m k .
Therefore, the homogeneous coordinate ring of π −1 2 (y) is where J is a satured ideal of R depending on y. Let y ∈ Y m−1 . As dim π −1 (y) = m − 1, one has dim(R I ⊗ B k(y)) = dim R/J = m. Proof. Let y = (p 0 : p 1 : · · · : p n ) ∈ Y m−1 . Without loss of generality, we can assume that p 0 = 1. Hence, is the defining ideal of y. For any f ∈ B, we have It follows that f + p = v + p. This implies that B/p ≃ k As N p is an Artinian B p -module and dim k (B/p) s = dim k (k[T 0 ]) s = 1 for any s ≥ 0, one has It follows that In this section, we consider a parameterization φ : P 2 k P 3 k of surface defined by four homogeneous polynomials f 0 , . . . , f 3 ∈ R = k[X 0 , X 1 , X 2 ] of the same degree d such that gcd(f 0 , . . . , f 3 ) = 1. Denote the homogeneous maximal ideal of R by m = (X 0 , X 1 , X 2 ).
From now on we assume that B is locally a complete intersection. Under this hypothesis, the module K in the exact sequence . . , f 3 with coefficients in R, respectively. Since the ideal I = (f) is homogeneous, these modules inherit a natural structure of graded R-modules. Let Z • be the approximation complex associated to I. The approximation complexes were introduced by Herzog, Simis and Vasconcelos in [8] to study the Rees and symmetric algebras of ideals. By definition where v 1 (a 0 , a 1 , a 2 , a 3 ) = a 0 T 0 + · · · + a 3 T 3 . As B is locally a complete intersection, the complex (Z • ) is acyclic and is a resolution of H 0 (Z • ) ≃ S I , see [9,Theorem 4]. Hence, the delicate case is when the ideal I satisfies indeg(I sat ) = indeg(I) = d. In this case, the first author in [3] established an upper bound for n = dim k H 1 m (R/I) d−2 in terms of d as follows.