A relative Hilbert-Mumford criterion

We generalize the classical Hilbert-Mumford criteria for GIT (semi-)stability in terms of one parameter subgroups of a linearly reductive group G over a field k, to the relative situation of an equivariant, projective morphism X -> Spec A to a noetherian k-algebra A. We also extend the classical projectivity result for GIT quotients: the induced morphism X^ss/G -> Spec A^G is projective. As an example of applications to moduli problems, we consider degenerations of Hilbert schemes of points.Comment: v4: minor correction

Speech Analysis

Contains research objectives and reports on one research project.National Science Foundatio

Speech Analysis

Contains reports on one research project

The COMPLETE Survey of Outflows in Perseus

We present a study on the impact of molecular outflows in the Perseus molecular cloud complex using the COMPLETE survey large-scale 12CO(1-0) and 13CO(1-0) maps. We used three-dimensional isosurface models generated in RA-DEC-Velocity space to visualize the maps. This rendering of the molecular line data allowed for a rapid and efficient way to search for molecular outflows over a large (~ 16 sq. deg.) area. Our outflow-searching technique detected previously known molecular outflows as well as new candidate outflows. Most of these new outflow-related high-velocity features lie in regions that have been poorly studied before. These new outflow candidates more than double the amount of outflow mass, momentum, and kinetic energy in the Perseus cloud complex. Our results indicate that outflows have significant impact on the environment immediately surrounding localized regions of active star formation, but lack the energy needed to feed the observed turbulence in the entire Perseus complex. This implies that other energy sources, in addition to protostellar outflows, are responsible for turbulence on a global cloud scale in Perseus. We studied the impact of outflows in six regions with active star formation within Perseus of sizes in the range of 1 to 4 pc. We find that outflows have enough power to maintain the turbulence in these regions and enough momentum to disperse and unbind some mass from them. We found no correlation between outflow strength and star formation efficiency for the six different regions we studied, contrary to results of recent numerical simulations. The low fraction of gas that potentially could be ejected due to outflows suggests that additional mechanisms other than cloud dispersal by outflows are needed to explain low star formation efficiencies in clusters.Comment: Published in The Astrophysical Journa

Speech Analysis

Contains reports on two research projects

Speech Analysis

Contains reports on two research projects

A GIT construction of degenerations of Hilbert schemes of points

We present a Geometric Invariant Theory (GIT) con-struction which allows us to construct good projective degenerationsof Hilbert schemes of points for simple degenerations. A comparisonwith the construction of Li and Wu shows that our GIT stack andthestack they construct are isomorphic, as are the associated coarse mod-uli schemes. Our construction is sufficiently explicit to obtain goodcontrol over the geometry of the singular fibres. We illustrate thisby giving a concrete description of degenerations of degreenHilbertschemes of a simple degeneration with two components

The geometry of degenerations of Hilbert schemes of points

Given a strict simple degeneration $f \colon X\to C$ the first three authors previously constructed a degeneration $I^n_{X/C} \to C$ of the relative degree $n$ Hilbert scheme of $0$-dimensional subschemes. In this paper we investigate the geometry of this degeneration, in particular when the fibre dimension of $f$ is at most $2$. In this case we show that $I^n_{X/C} \to C$ is a dlt model. This is even a good minimal dlt model if $f \colon X \to C$ has this property. We compute the dual complex of the central fibre $(I^n_{X/C})_0$ and relate this to the essential skeleton of the generic fibre. For a type II degeneration of $K3$ surfaces we show that the stack ${\mathcal I}^n_{X/C} \to C$ carries a nowhere degenerate relative logarithmic $2$-form. Finally we discuss the relationship of our degeneration with the constructions of Nagai.Comment: 53 pages. To appear in J. Algebraic Geo

Speech Analysis

Contains reports on two research projects.National Science Foundatio