David Eisenbud

Linear Free Resolutions and Minimal Multiplicity

Let S = k[x, ,..., x,] be a polynomial ring over a field and let A4 = @,*-a, M, be a finitely generated graded module; in the most interesting case A4 is an ideal of S. For a given natural number p, there is a great interest in the question: Can M be generated by (homogeneous) elements of degree <p? No simple answer, say in terms of the local cohomology of M, is known; but somewhat surprisingly the stronger question: Can the jth syzygy of M be generated by elements of degree Qp + j for all j = 0, l,..., n? does admit a simple response. The following is neither new nor difficult to prove, though it seems not well known:

Young diagrams and determinantal varieties

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Limit linear series: Basic theory.

AbstractIn this paper we introduce techniques for handling the degeneration of linear series on smooth curves as the curves degenerate to a certain type of reducible curves, curves of compact type. The technically much simpler special case of 1-dimensional series was developed by Beauville [2], Knudsen [21–23], Harris and Mumford [17], in the guise of “admissible covers”. It has proved very useful for studying the Moduli space of curves (the above papers and Harris [16]) and the simplest sorts of Weierstrass points (Diaz [4]). With our extended tools we are able to prove, for example, that:1)The Moduli spaceMg of curves of genusg has general type forg≧24, and has Kodaira dimension ≧1 forg=23, extending and simplifying the work of Harris and Mumford [17] and Harris [16].2)Given a Weierstrass semigroup Γ of genusg and weightw≦g/2 (and in a somewhat more general case) there exists at least one component of the subvariety ofMg of curves possessing a Weierstrass point of semigroup Γ which has the “expected” dimension 3g-2−w (and in particular, this set is not empty).3)The fundamental group of the space of smooth genusg curves having distinct “ordinary” Weierstrass points acts on the Weierstrass points by monodromy as the full symmetric group.4)Ifr andd are chosen so that

Generic Free Resolutions and a Family of Generically Perfect Ideals

\rho : = g - (r + 1)(g - d + r) = 0,

Existence, decomposition, and limits of certain Weierstrass points.

then the general curve of genusg has a certain finite number ofgdr’s [15, 20]. We show that the family of all these, allowing the curve to vary among general curves, is irreducible, so that the monodromy of this family acts transitively. If4=1, we show further that the monodromy acts as the full symmetric group.5)Ifr andd are chosen so that

Every algebraic set inn-space is the intersection ofn hypersurfaces

\rho = - 1,