1 Hilbert modular surfaces 3 1.1 The Hilbert modular group . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 The Baily-Borel compactification . . . . . . . . . . . . . . . . . . . . . . . 6 1.2.1 Siegel domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.3 Hilbert modular forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.4 Mk(Γ) is finite dimensional...

The toric Hilbert scheme of a lattice L ⊆ Z is the multigraded Hilbert scheme parameterizing all ideals in k[x1, . . . , xn] with Hilbert function value one for every g in the grading monoid G = N/L. In this paper we show that if L is twodimensional, then the toric Hilbert scheme of L is smooth and irreducible. This result is false for lattices of dimension three and higher as the toric Hilbert...

Steingrı́msson (2001) showed that the chromatic polynomial of a graph is the Hilbert function of a relative StanleyReisner ideal. We approach this result from the point of view of Ehrhart theory and give a sufficient criterion for when the Ehrhart polynomial of a given relative polytopal complex is a Hilbert function in Steingrı́msson’s sense. We use this result to establish that the modular and ...

The toric Hilbert scheme of a lattice L ⊆ Z is the multigraded Hilbert scheme parameterizing all ideals in k[x1, . . . , xn] with Hilbert function value one for every g in the grading monoid G = N/L. In this paper we show that if L is twodimensional, then the toric Hilbert scheme of L is smooth and irreducible. This result is false for lattices of dimension three and higher as the toric Hilbert...

The Hilbert scheme H n of n points in A contains an irreducible component R n which generically represents n distinct points in A. We show that when n is at most 8, the Hilbert scheme H n is reducible if and only if n = 8 and d ≥ 4. In the simplest case of reducibility, the component R 8 ⊂ H 8 is defined by a single explicit equation which serves as a criterion for deciding whether a given idea...

In this paper, those splendid characters of the Hilbert transform let the processes that taking wavelet transform after taking Hilbert transform for the statistic self-similarity processes FBM [ ) (t BH ] become another processes, that firstly taking Hilbert transform for the wavelet function ) (t φ and forming a new wavelet function ) (t ψ , secondly taking the wavelet transform for ) (t BH . ...

We consider the following open questions. Fix a Hilbert function h, that occurs for a reduced zero-dimensional subscheme of P. Among all subschemes, X, with Hilbert function h, what are the possible Hilbert functions and graded Betti numbers for the first infinitesimal neighborhood, Z, of X (i.e. the double point scheme supported on X)? Is there a minimum (h) and maximum (h) such function? The ...