Algorithms to Compute Chern-Schwartz-Macpherson and Segre Classes and the Euler Characteristic

Transcription

1 Algorithms to Compute Chern-Schwartz-Macpherson and Segre Classes and the Euler Characteristic Martin Helmer University of Western Ontario Abstract Let V be a closed subscheme of a projective space P n. We give algorithms to compute the Chern-Schwartz-MacPherson class, Euler characteristic and Segre class of V. These algorithms can be implemented using either symbolic or numerical methods. The basis for these algorithms is a method for calculating the projective degrees of a rational map defined by a homogeneous ideal. When combined with formula for the Chern-Schwartz-MacPherson class of a projective hypersurface and the Segre class of a projective variety in terms of the projective degrees of certain rational maps this gives us algorithms to compute the Chern-Schwartz- MacPherson class and Segre class of a projective variety. Since the Euler characteristic of V is the degree of the zero dimensional component of the Chern-Schwartz-MacPherson class of V our algorithm also computes the Euler characteristic χ(v ). The algorithms are tested on several examples and are found to perform favourably compared to other algorithms for computing Chern-Schwartz-MacPherson classes, Segre classes and Euler characteristics. Keywords: Euler characteristic, Chern-Schwartz-MacPherson class, Segre class, computer algebra, computational intersection theory. 1 Introduction Throughout this note we let V be a closed subscheme of P n = Proj(k[x 0,..., x n ]), with k an algebraical closed field of characteristic zero. Also let A (P n ) = Z[h]/(h n+1 ) denote the Chow ring of P n where h = c 1 (O P n(1)) is the class of a hyperplane in P n. Below we describe an algorithm originally given by the author of this note in [10]. This algorithm computes the Segre class, Chern-Schwartz-MacPherson class and the Euler characteristic of a projective variety over an algebraically closed field of characteristic zero. In particular, given the ideal I defining a projective variety V in P n we will compute the pushforward to P n of both the Segre class of V in P n and the Chern-Schwartz-MacPherson class of V (we abuse notation and denote the pushforwards to P n as s(v, P n ) and c SM (V ) respectively). From c SM (V ) we may immediately obtain the Euler characteristic of V, χ(v ) using the well-known relation which states that χ(v ) is equal to the degree of the zero dimensional component of c SM (V ). The algorithm described may be implemented either symbolically, with the computations relying on Gröbner bases calculations, or numerically using homotopy continuation. The topological Euler characteristic χ provides a useful and interesting invariant for the study of a wide variety of topics, both in mathematics and in applications. In addition to being an important topological invariant, the Euler characteristic has diverse applications, for example it is applied to problems of maximum likelihood estimation in algebraic statistics by Huh [11] and to string theory in physics by Collinucci et al. [5] and by Aluffi and Esole [3]. The desire to compute c SM classes explicitly is motivated partially by the relation to the Euler characteristic. In addition to containing the Euler characteristic c SM classes are an important invariant in algebraic geometry, providing a generalization of the Chern class to singular schemes and also having desirable functorial properties and relationships to other common invariants. The Chern-Schwartz-MacPherson class has also been directly related to problems in string theory in Aluffi and Esole [2]. Let V be an arbitrary subscheme of P n defined by a homogeneous ideal I = (f 0,..., f r ). The first algorithm to compute c SM (V ) was that of Aluffi [1]. For a hypersurface V, this algorithm

2 requires the computation of the blowup of P n along the singularity subscheme of V. The computation of such blowups can be an expensive operation, making this algorithm impractical for many examples. Another algorithm to compute the c SM class of a hypersurface was given by Jost in [12]. This method works by computing the degrees of certain residual sets via a particular saturation. The algorithm described here uses a result originally proved by the author of this note in Theorem 3.1 of [10], which is given below as Theorem 2.1, combined with a result of Aluffi [1], given in (4) below. Specifically, the result of Theorem 2.1 gives a formula to compute the projective degrees of the rational map φ : P n P m associated to the homogeneous ideal (f 0,..., f m ) k[x 0,..., x n ]. This result is used to construct an algorithm which computes the projective degrees of a rational map and can be implemented in most computer algebra systems either symbolically with standard Gröbner bases algorithms, or numerically using homotopy continuation. The result of Aluffi [1] then allows us to directly find the c SM class of a hypersurface from certain projective degrees. This method of computing the c SM class using projective degrees provides a performance improvement in many cases, see Table 1. All these methods require the use of the inclusion-exclusion property of c SM classes when V has codimension higher than one. Specifically for V 1, V 2 subschemes of P n the inclusion-exclusion property for c SM classes states c SM (V 1 V 2 ) = c SM (V 1 ) + c SM (V 2 ) c SM (V 1 V 2 ). (1) Note that this relation for c SM classes will allow us to reduce all computation of c SM classes to the case of hypersurfaces. In the case of subvarieties of P n we have the following proposition which follows directly from (1). Proposition 1. Let V = X 1 X r = V (f 1 ) V (f r ) be a subscheme of P n = Proj(k[x 0,..., x n ]). Write the polynomials defining V as F = (f 1,..., f r ) and let F {S} = i S f i for S {1,..., r}. Then, c SM (V ) = ( 1) S +1 ( c SM V (F{S} ) ) S {1,...,r} where S denotes the cardinality of the integer set S. Note that the use of inclusion/exclusion requires exponentially many c SM computations relative to the number of generators of I. Additionally we must consider c SM classes of products of many or all of the generators of I, which may have significantly higher degree than the original scheme V. The algorithm of Aluffi [1] is also capable of computing Segre classes. As with Aluffi s algorithm for c SM classes this algorithm works by computing blow-ups of P n along certain subvarieites; hence the main cost of Aluffi s algorithm is the cost of computing Rees algebras, which can be quite computationally expensive in many cases. Another algorithm to compute Segre classes is the algorithm of Eklund, Jost and Peterson [7], which works by finding the degree of a certain residual set via a particular saturation. Let J be a homogeneous ideal defining a scheme Y P n. The algorithm to compute Segre classes described in this note works by computing the projective degree of a certain rational map defined by the ideal J using Theorem 2.1, and then applying the result of Aluffi [1] given in (5). As with c SM classes this algorithm using projective degrees to compute Segre classes provides improved performance relative to other algorithms for many examples, see Table 2. 2 A Theorem on the Projective Degree Consider a rational map φ : P n P m. In the manner of Harris (Example 19.4 of [9]) we may define the projective degrees of the map φ as a list of integers (g 0,..., g n ) where g i = card ( φ 1 ( P m i) P i). (2) where P m i P m and P i P n are general hyperplanes of dimension m i and i respectively and card is the cardinality of a zero dimensional set. Note that points in ( φ 1 ( P m i) P i) will have multiplicity one (this follows from the Bertini theorem of Sommese and Wampler [13, A.8.7]). The result below will allow us to compute the projective degrees of a rational map specified by a homogeneous ideal using a computer algebra system. This result is proved by the author of this note in Theorem 3.1 of [10].

3 Theorem 2.1 (Theorem 3.1 Helmer [10]). Let I = (f 0,..., f m ) be a homogeneous ideal in R = k[x 0,..., x n ] defining a ϱ-dimensional scheme V = V (I), and assume, without loss of generality, that all the polynomials f i generating I have the same degree. The projective degrees (g 0,..., g n ) of φ : P n P m, φ : p (f 0 (p) : : f m (p)), are given by g i = dim k (R[T ]/(P P i + L L n i + L A + S)). ( m ) Here P l, L l, L A and S are ideals in R[T ] = k[x 0,..., x n, T ] with P l = j=0 λ l,jf j for λ l,j a general scalar in k, S = 1 T m ( j=0 ϑ jf j ), for ϑ j a general scalar in k, L l a general homogeneous linear form for l = 1,..., n and L A a general affine linear form. Additionally g 0 = 1. 3 An Algorithm to Compute the c SM class and Euler Characteristic Let V be a hypersurface of P n defined by the homogeneous polynomial ideal (f) in k[x 0,..., x n ]. Using the partial derivatives of f we define a rational map ϕ : P n P n, ( ) f f ϕ : p (p) : : (p). (3) x 0 x n This map is referred to as the polar map or gradient map [6]. Let V = V (f) and let (g 0,..., g n ) be the projective degrees of the polar map corresponding to f (3), by Theorem 2.1 of Aluffi [1] we have the following equality in A (P n ) = Z[h]/h n+1 c SM (V ) = (1 + h) n+1 n g j ( h) j (1 + h) n j. (4) This result of (4) combined with the inclusion/exclusion property of c SM classes and the result of Theorem 2.1 gives us an algorithm to compute c SM (V ) for an arbitrary scheme V = V (I) in P n. We summarize this algorithm below. j=0 Input: A homogeneous ideal (f 0,..., f r ) in k[x 0,..., x n ] defining a scheme V = V (I) in P n. Output: c SM (V ) and/or χ(v ). Make a list L I of all generators and all products of generators of the ideal I. For each polynomial f L I compute c SM (V (f)). Compute the projective degrees (g 0,..., g n ) of the polar map of f (3) using Theorem 2.1. Compute c SM (V (f)) from the projective degree (g 0,..., g n ) using (4). Apply the inclusion/exclusion property of c SM classes as in Proposition 1 to obtain c SM (V ). In Table 1 we compare the running times of several algorithms to compute the c SM class and Euler characteristic. We note that in all cases considered here our algorithm using Theorem 2.1 to compute the projective degree performs favourably in comparison to other known algorithms which compute the c SM class and Euler Characteristic. Test computations for the symbolic version were performed over GF(32749), yielding the same results found by working over Q for all examples considered. The symbolic methods are somewhat slower when performed over Q, but still much faster than the numeric ones. 4 An Algorithm to Compute the Segre Class Consider a subscheme Y = V (J) P n defined by a homogeneous ideal J = (w 0,..., w m ) R = k[x 0,..., x n ]. Assume, without loss of generality, that deg(w i ) = d for all i. Also let (g 0,..., g n )

4 Input CSM (Aluffi) CSM (Jost) csm polar (M2) csm polar (Sage) euler Twisted cubic 0.3s 0.1s (35s) 0.1s (37s) 0.1s (0.6s) 0.2s Segre embedding of P 1 P 2 in P 5 0.4s 0.8s (148s) 0.2s (152s) 0.2s (57s) 0.2s Smooth degree 3 surface in P 4-1.2s (-) 0.6s (-) 0.2s (28s) 20.1s Smooth degree 7 surface in P 4-50s 42s 7.9s - Smooth degree 3 surface in P s 2.2s 1.0s - Smooth degree 3 surface in P s 2.0s - Sing. degree 3 surface in P s 2.3s n/a Deg. 12 hypersurface in P s 1.0s 0.2s 0.1s n/a Sing. Var. in P s 0.3s n/a Table 1: Our algorithm to compute the c SM class and Euler characteristic of an arbitrary subscheme of P n using Theorem 2.1 combined with (4) and inclusion/exclusion is denoted csm polar. All algorithms present here use inclusion/exclusion, all algorithms are implemented in Macaulay2 (M2); csm polar is also implemented in Sage [14]. Euler is the built in M2 routine for computing Euler characteristics using Hodge numbers, it is only valid for non-singular schemes. CSM (Jost) is described in [12] and computes the degrees of certain residual sets using a saturation. CSM (Aluffi) is described in [1] and requires the computation of a certain Rees algebra. Timings in () are for the numerical versions, using either Bertini [4] (via M2) or PHCpack [15] (via Sage). Computations that do not finish within 600s are denoted -, or are omitted for numerical versions. be the projective degrees of the rational map φ : P n P m, φ : p (w 0 (p) : : w m (p)). By Proposition 3.1 of Aluffi [1] we have s(y, P n ) = 1 n i=0 g i h i (1 + dh) i+1 A (P n ). (5) This result combined with Theorem 2.1 gives us an algorithm to compute the Segre class of a scheme Y = V (J) P n. We summarize this algorithm below. Input: A homogeneous ideal J = (w 0,..., w m ) in k[x 0,..., x n ] defining a scheme Y = V (J) in P n. Output: The Segre class s(y, P n ). Let φ be the rational map φ : P n P m, φ : p (w 0 (p) : : w m (p)). Compute the projective degrees (g 0,..., g n ) of the rational map φ using Theorem 2.1. Compute s(y, P n ) using (5). In Table 2 we compare the running times of several algorithms to compute the Segre class of a projective scheme specified by a homogeneous ideal. We note that in all cases considered here our algorithm using Theorem 2.1 to compute the projective degree performs favourably in comparison to other known algorithms which compute these Segre classes. Test computations for the symbolic version were performed over GF(32749), yielding the same results found by working over Q for all examples considered. The symbolic methods are somewhat slower when performed over Q, but still much faster than the numeric ones. Input Segre(Aluffi [1]) segreclass(e.j.p. [7]) segre proj deg(theorem 1) Rational normal curve in P 7-7s (9s) 8s (15s) Smooth surface in P 8 defined by minors - 59s 18s Degree 48 surface in P 6-172s 6s Singular var. in P s 33s 10s Singular var. in P 6-173s 6s Segre embedding of P 2 P 3 in P 11 2s - 52s Table 2: Our algorithm using (5) and Theorem 2.1 is denoted segre proj deg. Segre(Aluffi [1]) requires the computation of a certain Rees algebra, segreclass(e.j.p. [7]) requires the computation of the degrees of certain residual sets using a saturation. All algorithms are implemented in Macaulay2 [8]. Timings in () are for the numerical versions, using either Bertini [4] (via M2) or PHCpack [15] (via Sage). Computations that do not finish within 600s are denoted -. Numerical timings which did not finish in 600s are omitted.

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