A pr 2 00 9 Abel maps for curves of compact type

Naturality of Abel maps

We give a combinatorial characterization of nodal curves admitting a natural (i.e. compatible with and independent of specialization) d-th Abel map for any d ≥ 1. Let X be a smooth projective curve and d a positive integer; the classical d-th Abel map of X , αX : X d −→ PicX , associates to (p1, . . . , pd) ∈ X the class of the line bundle OX(p1 + . . .+ pd) in Pic d X . Such a morphism has goo...

M ay 2 00 9 A De Vries - type Duality Theorem for Locally Compact Spaces – I ∗

A duality theorem for the category of locally compact Hausdorff spaces and continuous maps which generalizes the well-known Duality Theorem of de Vries is proved. 2000 MSC: primary 18A40, 54D45; secondary 06E15, 54C10, 54E05, 06E10.

m at h . A G ] 1 9 A pr 2 00 6 ON RATIONAL CUSPIDAL CURVES , OPEN SURFACES AND LOCAL SINGULARITIES

ul 2 00 5 Set maps , umbral calculus , and the chromatic

Some important properties of the chromatic polynomial also hold for any polynomial set map satisfying pS(x + y) = ∑ T⊎U=S pT (x)pU (y). Using umbral calculus, we give a formula for the expansion of such a set map in terms of any polynomial sequence of binomial type. This leads to several new expansions of the chromatic polynomial. We also describe a set map generalization of Abel polynomials.

Limit Linear Series: Constructions and Applications

We discuss the Eisenbud-Harris theory of limit linear series, and survey a number of applications, ranging from the geometry of moduli spaces of curves to connectedness of certain Hurwitz spaces. We also present a more recent construction of limit linear series, and discuss applications and potential applications both to classical linear series and to their higher-rank generalization. In this s...