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Star configurations of points are configurations with known (and conjectured) extremal behaviors among all configurations of points in $\mathbb P_k^n$; additional interest come from their rich structure, which allows them to be studied using tools from algebraic geometry, combinatorics, commutative algebra and representation theory. In the present paper we investigate the more general problem of determining the structure of symbolic powers of a wide generalization of star configurations of points (introduced by Geramita, Harbourne, Migliore and Nagel) called star configurations of hypersurfaces in $\mathbb P_k^n$. Here (1) we provide explicit minimal generating sets of the symbolic powers $I^{(m)}$ of these ideals $I$, (2) we introduce a notion of $\delta$-c.i. quotients, which generalize ideals with linear quotients, and show that $I^{(m)}$ have $\delta$-c.i. quotients, (3) we show that the shape of the Betti tables of these symbolic powers is determined by certain "Koszul" strands and we prove that a little bit more than the bottom half of the Betti table has a regular, almost hypnotic, pattern, and (4) we provide a closed formula for all the graded Betti numbers in these strands. As a special case of (2) we deduce that symbolic powers of ideals of star configurations of points have linear quotients. We also improve and extend results by Galetto, Geramita, Shin and Van Tuyl, and provide explicit new general formulas for the minimal number of generators and the symbolic defects of star configurations. Finally, inspired by Young tableaux, we introduce a technical tool which may be of independent interest: it is a "canonical" way of writing any monomial in any given set of polynomials. Our methods are characteristic--free., Comment: Final revision (original paper was accepted for publication in Trans. Amer. Math. Soc.)

Let ( R , m ) (R,\mathfrak {m}) be an n n -dimensional regular local ring, essentially of finite type over a field of characteristic zero. Given an m \mathfrak {m} -primary ideal a \mathfrak {a} of R R , the relationship between the singularities of the scheme defined by a \mathfrak {a} and those defined by the multiplier ideals J ( a c ) \mathcal {J}(\mathfrak {a}^c) , with c c varying in Q + \mathbb {Q}_+ , are quantified in this paper by showing that the Samuel multiplicity of a \mathfrak {a} satisfies e ( a ) ≥ ( n + k ) n / c n e(\mathfrak {a}) \ge (n+k)^n/c^n whenever J ( a c ) ⊆ m k + 1 \mathcal {J}(\mathfrak {a}^c) \subseteq \mathfrak {m}^{k+1} . This formula generalizes an inequality on log canonical thresholds previously obtained by Ein, Mustaţǎ and the author of this paper. A refined inequality is also shown to hold for small dimensions, and similar results valid for a generalization of test ideals in positive characteristics are presented.

Classical invariants for representations of one Lie group can often be related to invariants of some other Lie group. Physics suggests that the right objects to consider for these questions are certain refinements of classical invariants known as conformal blocks. Conformal blocks appear in algebraic geometry as spaces of global sections of line bundles on the moduli stack of parabolic bundles on a smooth curve. Rank-level duality connects a conformal block associated to one Lie algebra to a conformal block for a different Lie algebra. In this paper we prove a rank-level duality for type so(2r+1) on the pointed projective line conjectured by T. Nakanishi and A. Tsuchiya.

Generic reducing fields of Jordan pairs, generalizing at the same time generic splitting fields of associative algebras and generic zero fields of quadratic forms, are intrinsically defined and constructed. The most elementary properties are derived, and the relationship with other generic constructions, particularly those linked to Brauer-Severi varieties, are investigated. As an application it is shown that there exist nonisomorphic exceptional Jordan division algebras having the same splitting fields. After their first, though somewhat implicit, appearance in a paper by Witt [28], generic splitting fields of algebraic structures received extensive treatment in the setting of associative algebras by Amitsur [1] and Roquette [23, 24]. Having been investigated further by Ferrar [5] for composition algebras and Knebusch [11, 12] for quadratic forms, the latter's point of view leading to new perspectives in the associative theory pursued by Heuser [7], Kovacs [13], and Saltman [25], Jacobson [8], appealing to the theory of Jordan algebras, recently made contributions along these lines to associative algebras with involution. In the present paper, we go one step further and propose the theory of Jordan pairs as a general framework where a fair amount of the material mentioned above may be recovered in a unified fashion. It goes without saying that the sole justification of such an approach lies in a treatment which is intrinsic, i.e., which does not require the classification of finite-dimensional simple Jordan pairs. This is accomplished here while, at the same time, keeping the amount of "abstract" Jordan pair theory to comparatively modest proportions. Among the various peculiarities which are typical for the point of view adopted here, the following is particularly important: In dealing with finite-dimensional Jordan structures, one has to distinguish carefully between the property of being reduced as compared to the one of being split. In fact, the former notion obviously being the more elementary of the two, it is clear that a first thrust into the subject must concern itself with reducing fields rather than splitting fields of Jordan pairs. As a rule, all generic splitting fields known to date may conveniently be interpreted as rational function fields of appropriate algebraic varieties. The present paper provides no exceptions. In fact, we find it convenient to deduce a number of properties for generic reducing fields by making use of this interpretation, blaming everything on elementary results from algebraic geometry. To be more explicit, after having derived the most elementary properties in ?2, we construct two classes of generic reducing fields, the first one being related, in the manner described above, Received by the editors November 21, 1983. 1980 Mathematics Suject Classfication. Primary 17C20; Secondary 1OC10, 12F20, 16A46. (@)1984 American Mathematical Society 0002-9947/84 $1.00 + $.25 per page 825 This content downloaded from 207.46.13.180 on Mon, 26 Sep 2016 06:14:10 UTC All use subject to http://about.jstor.org/terms

The main results of this paper are stated in 2.8 and 3.7; the notations and language are those used by the author in his previous work; numbers in brackets refer to the bibliography at the end of the paper. Although a few results of ??1 and 2 are valid for any characteristic, this paper intends to treat only the case of characteristic zero. The case of positive characteristic is treated in a forthcoming paper, since it needs an analysis of the derivations of higher order, and is connected to the rather surprising fact that an abelian variety over a field of positive characteristic may very well possess exact differentials of the first kind. ??1 and 2 contain the proofs of those properties of the differentials of the second kind which are needed in ?3; these properties are familiar in classical algebraic geometry, but their algebraic proofs are new. While our main interest, in ??1 and 2, rests with the differentials of the second kind, certain properties of the differentials of the first kind are also found, usually as special cases; the algebraic proofs of most of these properties are not new, and can be found also in [S. Koizumi, On the differentialforms of the first kind on algebraic varieties, J. Math. Soc. Japan vol. 1 (1949) p. 273] and [S. Nakano, On invariant differential forms on group varieties, ibid. vol. 2 (1951) p. 216]. 1. The differentials of the first and second kind. Let V be an irreducible variety over the field k; let W be an irreducible subvariety of V, and set Q=Q(W/V), $1= $3(W/V); let D be a derivation on V (see ?5 of [4]). We shall say that D is regular at W if Dx E Q when x E Q, and Dx (=-3 when x (= . If, in addition, DxEG when xCQ, we shall say that D has a zero at W. Let '7r be the homomorphic mapping of Q into k(W) whose kernel is 3; if D is regular at W, for xGQ we see that '7r(Dx) depends only on '7rx, and from this follows the existence of a derivation 7rD on W such that (7rD) ('7rx) = '7r(Dx) for x E Q; rD is called the derivation induced by D on W; clearly, 7rD=0 if and only if D has a zero at W.

1. In a previous paper [10, hereafter cited DGHO] the author studied the osculating spaces of submanifolds of affine and projective spaces. This led to a theory of affine and projective singularities (e.g., inflection points, planar points) which were described by the falling-in-rank of certain vector-bundle homomorphisms. This theory he then developed in the complex-analytic case and obtained generalizations of the Plucker formulas of algebraic geometry. In his thesis E. A. Feldman [5] developed the theory extensively for real differentiable manifolds and obtained transversality theorems and theorems on the nonexistence of singularity-free immersions. These theorems constitute nearly all that is known about real affine differential geometry in-the-large in higher dimensions. The main purpose of the present paper is to generalize the theory to submanifolds of affinely connected spaces, and following a suggestion of S. S. Chern, to submanifolds of projectively connected spaces. The main features of the affine case remain the same. To treat the projective case we shall show that every projective connexion has an underlying affine connexion and that the osculating spaces and inflection points defined with respect to this affine connexion depend only on the projective connexion. The geometry of spaces with projective connexion seems to be the natural setting for many of the main results of Feldman's thesis. In reality the generalization to affinely connected spaces has already been carried out by Feldman. He has developed the theory for a space with affine connexion together with a further structure which he calls a sequence of dissections or "pth order symmetric linear connexions." Our essential contribution is to show that the affine connexion already determines the additional structure in a natural way. As an example of the theory we shall state and prove one of Feldman's theorems. The main results of this paper are two theorems on connexions. Let M be a differentiable manifold and let Tp be the pth order osculating bundle of M, i.e., the bundle of linear differential operators of order ?p on functions on M. In DGHO the author established a natural exact sequence

The classification of normal singular cubic surfaces in P 3 \mathbf {P}^3 over a complex number field C \mathbf {C} was given by J. W. Bruce and C. T. C. Wall. In this paper, first we prove their results by a different way, second we provide normal forms of normal singular cubic surfaces according to the type of singularities, and finally we determine automorphism groups on normal singular cubic surfaces with no parameters.

If L ( a , s ) := ∑ n c ( n , a ) n − s L(a,s):=\sum _n c(n,a)n^{-s} is a family of “geometric” L − L- functions depending on a parameter a a , then the function ( p , a ) ↦ c ( p , a ) (p,a)\mapsto c(p,a) , where p p runs through the set of prime integers, is not a rational function and hence is not a function belonging to algebraic geometry. The aim of the paper is to show that if one enlarges algebraic geometry by “adjoining a Fermat quotient operation”, then the functions c ( p , a ) c(p,a) become functions in the enlarged geometry at least for L − L- functions of curves and Abelian varieties.

This article concerns an old conjecture due to E. T. Whittaker, aiming to describe the group uniformizing an arbitrary hyperelliptic Riemann surface y 2 = ∏ i = 1 2 g + 2 ( x − a i ) y^2=\prod _{i=1}^{2g+2}(x-a_i) as an index two subgroup of the monodromy group of an explicit second order linear differential equation with singularities at the values a i a_i . Whittaker and collaborators in the thirties, and R. Rankin some twenty years later, were able to prove the conjecture for several families of hyperelliptic surfaces, characterized by the fact that they admit a large group of symmetries. However, general results of the analytic theory of moduli of Riemann surfaces, developed later, imply that Whittaker’s conjecture cannot be true in its full generality. Recently, numerical computations have shown that Whittaker’s prediction is incorrect for random surfaces, and in fact it has been conjectured that it only holds for the known cases of surfaces with a large group of automorphisms. The main goal of this paper is to prove that having many automorphisms is not a necessary condition for a surface to satisfy Whittaker’s conjecture.

Schubert varieties in the flag manifold SL(n)/B play a key role in our understanding of projective varieties. One important problem is to determine the locus of singular points in a variety. In 1990, Lakshmibai and Sandhya showed that the Schubert variety X w is nonsingular if and only if w avoids the patterns 4231 and 3412. They also gave a conjectural description of the singular locus of X w . In 1999, Gasharov proved one direction of their conjecture. In this paper we give an explicit combinatorial description of the irreducible components of the singular locus of the Schubert variety X w for any element w ∈ G n . In doing so, we prove both directions of the Lakshmibai-Sandhya conjecture. These irreducible components are indexed by permutations which differ from w by a cycle depending naturally on a 4231 or 3412 pattern in w. Our description of the irreducible components is computationally more efficient (O(n 6 )) than the previously best known algorithms, which were all exponential in time. Furthermore, we give simple formulas for calculating the Kazhdan-Lusztig polynomials at the maximum singular points.

In this paper we shall determine all actions of groups of prime order p with p > 5 on Gorenstein del Pezzo (singular) surfaces Y of Picard number 1. We show that every order-p element in Aut(Y) (= Aut(Y), Y being the minimal resolution of Y) is lifted from a projective transformation of P 2 . We also determine when Aut(Y) is finite in terms of K 2 Y , Sing Y and the number of singular members in |-K |. In particular, we show that either |Aut(Y)] = 2 a 3 b for some 1 5, there is at least one element gp of order p in Aut(Y) (hence |Aut(Y)| is infinite).

Let K K be the basic closed semi-algebraic set in R n \mathbb {R}^n defined by some finite set of polynomials S S and T T , the preordering generated by S S . For K K compact, f f a polynomial in n n variables nonnegative on K K and real ϵ > 0 \epsilon >0 , we have that f + ϵ ∈ T f+\epsilon \in T [15]. In particular, the K K -Moment Problem has a positive solution. In the present paper, we study the problem when K K is not compact. For n = 1 n=1 , we show that the K K -Moment Problem has a positive solution if and only if S S is the natural description of K K (see Section 1). For n ≥ 2 n\ge 2 , we show that the K K -Moment Problem fails if K K contains a cone of dimension 2. On the other hand, we show that if K K is a cylinder with compact base, then the following property holds: \[ ( ‡ ) ∀ f ∈ R [ X ] , f ≥ 0 on K ⇒ ∃ q ∈ T such that ∀ real ϵ > 0 , f + ϵ q ∈ T . (\ddagger )\quad \quad \forall f\in \mathbb {R}[X], f\ge 0 \text { on } K\Rightarrow \exists q\in T \text { such that }\forall \text { real } \epsilon >0, f+\epsilon q\in T.\quad \] This property is strictly weaker than the one given in [15], but in turn it implies a positive solution to the K K -Moment Problem. Using results of [9], we provide many (noncompact) examples in hypersurfaces for which ( ‡ \ddagger ) holds. Finally, we provide a list of 8 open problems.

We consider homogeneous ideals I and the initial ideal in(I) for the revlex order. First we give a sequence of invariants computed from I giving better and better "approximations" to the initial ideal and ending in an equivalent description. Then we apply this to different settings in algebraic geometry to understand what information is encoded in the generic initial ideal of the ideal of a projective scheme. We also consider the higher initial ideals as defined in a paper by Fl0ystad. In particular, we show that giving the generic higher initial ideal of a space curve is equivalent to giving the generic initial ideal of a linked curve.

Let Af be a compact C°° manifold. A theorem of Nash-Tognoli asserts that M has an algebraic model, that is, M is diffeomorphic to a nonsingular real algebraic set X. Let FV^AfX, Z/2) denote the subgroup of Hk(X, Z/2) of the cohomology classes determined by algebraic cycles of codi- mension k on X. Assuming that M is connected, orientable and dim M > 5 , we prove in this paper that a subgroup G of H2(M, Z/2) is isomorphic to H^ (X, Z/2) for some algebraic model X of M if and only if w2(TM) is in G and each element of G is of the form W2K) for some real vector bundle £ over M , where w2 stands for the second Stiefel-Whitney class. A result of this type was previously known for subgroups G of HX(M, Z/2).

Introduction. In classical algebraic geometry the adjoint curves to an irreducible plane curve are an essential tool in the study of the geometry on the curve. In this paper we shall give an algebro-arithmetic development of the theory of adjoint curves, and shall extend the classical results to irreducible plane curves with arbitrary singularities defined over arbitrary ground fields. Our definition of the adjoint condition at a given singular point of the curve is stated in terms of the conductor between the local ring of the point and its integral closure. The fundamental properties of the adjoint curves are then derived from corresponding properties of the conductor. The single deepest and most important property of the adjoint curves is that, on a curve of order m, the adjoint curves of order m -3 cut out the complete canonical series. This property is equivalent to the fact that the degree of the fixed component of the adjoint series is twice the number of conditions which the adjoint curves impose on the curves of sufficiently high order(1). We shall give two distinct and independent proofs of this proposition. The first proof is a direct one, based upon a detailed analysis of the singularities of the given curve. This analysis, to which part I is devoted, applies equally well to algebraic number fields, and our treatment will include this case with that of algebraic function fields of one variable. The second proof is more indirect, depending upon the Riemann-Roch theorem and a generalization of the classical representation theorem of the differentials of the first kind. This proof, which will be given in part II, holds only for plane curves whose function field is separably generated over the ground field.