Let S=k[x~ ..... x,] be a polynomial ring over an infinite field k, and let 1 be a homogeneous ideal of S.An algorithm for computing the (first) syzygies of I is due independently to Spear [Spe77] and Schreyer [Sch80]: One chooses an ordering on the monomials of S, and then constructs a monomial ideal in(1) generated by the lead terms of all elements of I, in(l) can be viewed as the limit of 1 under the action of a 1-parameter subgroup of GL(n) on the Hilbert scheme [Bay 82], so in(1) occurs as the special fiber of a flat family whose general fiber is isomorphic to I. It follows from a well-known criterion for flatness [Art 76] that each syzygy of in(l) can be lifted to a syzygy of I; the set of syzygies thus obtained can be trimmed to give a complete set of minimal syzygies of I.The rnonomial ideal in(I) was first studied The following problem arises in using this syzygy algorithm in practice: in(l) can have minimal generators and syzygies in degrees higher than any minimal generator or syzygy of I. In this situation, computations in these higher degrees are unnecessary; one should compute the generators and syzygies of in(1) in only those degrees necessary to find all minimal syzygies of 1.In order to modify the syzygy algorithm to take advantage of this observation, one would like a criterion for determining when all minimal syzygies of I have been found. This problem appears to be intractable at present. However, the question of bounding the degrees of the minimal jth syzygies of I, for all j, is tractable. Recall that I is defined to be m-regular if the jth syzygy module of I is generated in degrees 0 ([Mum66], [EiGo 84]).The regularity of I, reg(l), is defined to be the least m for which I is m-regular. We have reg(in (1) We give in w a criterion for 1 to be m-regular, which depends only on computations in the finite vector spaces S,, and S"+I of polynomials of degrees m, m+l:If one can find h 1 .... ,hieS 1, so that the subspaces ((I, h 1 ..... hi-1) : hi)m and (I, hi, ..., h i 1),. are equal for 1 s j, and (I, h 1 .... , hj)m=Sm, then I is m-regular. Furthermore, if I is m-regular, then a generic choice of h~, ..., hje S 1 will satisfy these conditions.One could use this result to terminate syzygy computations early, in cases where reg(in(I))>reg(I). However, further study reveals a close connection between this result and a particular order on the monomials of S, the reverse lexicographic order. This order is used to compute saturations in [Bay 82]. The reverse lexicographic order was then studied in characteristic zero, in generic coordinates, by several authors. It is observed in [Laz83] that under this hypothesis in low dimensions, the generators of in(I) are of particularly low degree. In [Giu84], this hypothesis is further studied, and a worst-case upper bound on the degrees of generators of in(l) is obtained, which improves Hermann's corresponding bound for ideal membership [Her 26]. In [Ang84], independent of a preliminary version of our results, it is shown that under this hypoth...

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We study the topological properties of Calabi-Yau threefold hypersurfaces at large h 1,1 . We obtain two million threefolds X by triangulating polytopes from the Kreuzer-Skarke list, including all polytopes with 240 ≤ h 1,1 ≤ 491. We show that the Kähler cone of X is very narrow at large h 1,1 , and as a consequence, control of the α expansion in string compactifications on X is correlated with the presence of ultralight axions. If every effective curve has volume ≥ 1 in string units, then the typical volumes of irreducible effective curves and divisors, and of X itself, scale as (h 1,1 ) p , with 3 p 7 depending on the type of cycle in question. Instantons from branes wrapping these cycles are thus highly suppressed.

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In this chapter we present an introduction to the structure of Macaulay 2 commands and the writing of functions in the Macaulay 2 language. For further details see the Macaulay 2 manual distributed with the program [1].

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