We construct a compactification $\mathcal{M}$_d of the moduli space of plane curves of degree d. We regard a plane curve C ⊂ℙ² as a surface-divisor pair (ℙ²,C), and we define $\mathcal{M}$_d as a moduli space of pairs (X,D), where X is a degeneration of the plane. We show that, if d is not divisible by 3, the stack $\mathcal{M}$_d is smooth and the degenerate surfaces X can be described explicitly.

For a finite subgroup G⊂SL(3,ℂ), Bridgeland, King, and Reid [BKR] proved that the moduli space of G-clusters is a crepant resolution of the quotient ℂ³/G . This paper considers the moduli spaces $\mathcal{M}$_θ, introduced by Kronheimer and further studied by Sardo Infirri, which coincide with G-Hilb for a particular choice of geometric invariant theory (GIT) parameter θ. For G Abelian, we prove that every projective crepant resolution of ℂ³/G is isomorphic to $\mathcal{M}$_θ for some parameter θ. The key step is the description of GIT chambers in terms of the K-theory of the moduli space via the appropriate Fourier-Mukai transform. We also uncover explicit equivalences between the derived categories of moduli $\mathcal{M}$_θ for parameters lying in adjacent GIT chambers.

We investigate the Anderson metal-insulator transition for random Schrödinger operators. We define the strong insulator region to be the part of the spectrum where the random operator exhibits strong dynamical localization in the Hilbert-Schmidt norm. We introduce a local transport exponent β(E) and set the weak metallic transport region to be the part of the spectrum with nontrivial transport (i.e., β(E)>0). We prove that these insulator and metallic regions are complementary sets in the spectrum of the random operator and that the local transport exponent β(E) provides a characterization of the metal-insulator transport transition. Moreover, we show that if there is such a transition, then β(E) has to be discontinuous at a transport mobility edge. More precisely, we show that if the transport is nontrivial, then β(E)≥1/(2d), where d is the space dimension. These results follow from a proof that slow transport of quantum waves in random media implies the starting hypothesis for the authors' bootstrap multiscale analysis. We also conclude that the strong insulator region coincides with the part of the spectrum where we can perform a bootstrap multiscale analysis, proving that the multiscale analysis is valid all the way up to a transport mobility edge.

In this paper we give explicit lower bounds for an integer m to be represented by a positive definite integral quadratic form Q in n≥3 variables defined over ℚ. As an example, we apply these bounds to answer affirmatively the long-standing conjecture of Kneser that the only positive integers not represented by x²+3y²+5z²+7w² are 2 and 22.

When n=3, the existence of spinor square classes and the possible existence of a Siegel zero complicates the estimate and requires us to restrict m to a finite union of square classes in order to obtain explicit constants. In this setting, we obtain a lower bound and asymptotics for the number of representations of m by Q, even within a spinor square class.

These methods can be easily generalized to obtain similar results for the representability of integers by a totally definite quadratic form over a totally real number field, and we carry out our local analysis in this generality. We also describe how to generalize these results to handle congruence conditions and representability by a rational quadratic polynomial.

We compute the Bott-Chern classes of the metrized relative Euler sequence describing the relative tangent bundle of the variety ℙ(E) of the hyperplanes in a holomorphic Hermitian vector bundle (E,h) on a complex manifold. We give applications to the construction of the arithmetic characteristic classes of an arithmetic vector bundle $\overline{\mathscr{E}}$ and to the computation of the height of ℙ($\overline{\mathscr{E}}$) with respect to the tautological quotient bundle $\mathscr{O}_\overline{\mathscr{E}}$(1).