The space of Cohen-Macaulay curves

One can consider the Hilbert scheme as a natural compactification of the space of smooth projective curves with fixed Hilbert polynomial. Here we consider a different modular compactification, namely the functor CM parameterizing curves together with a finite map to $\mathbb{P}^n$ that is generically a closed immersion. We prove that CM is an algebraic space by contructing a scheme W and a representable, surjective and smooth map W -> CM. Moreover, we show that CM satisfies the valuative criterion for properness.Comment: 39 page

Copine III interacts with ERBB2 and promotes tumor cell migration

Breast cancer is the most prevalent form of cancer in females: one of nine women develops breast cancer during her lifetime and it is predicted that one in 27 women will die as a result of this disease. Moreover, it is anticipated that with almost 30 % of females affected, breast cancer will be the most frequently diagnosed cancer in 2009 (www.cancer.org). Given these facts, much time and resources have been provided to research in the breast cancer area. The ErbB2 receptor tyrosine kinase is one of the most-studied oncogenes in breast cancer as amplification and overexpression of the ERBB2 gene is known to occur in up to 25 % of all affected patients and is correlated with a highly aggressive disease and poor patient prognosis. Our study focused on signaling molecules interacting with the C-terminal regulatory region of the ErbB2 receptor. We used T47D breast cancer cells metabolically labeled with SILAC to identify binding partners of the pTyr1248 site of ErbB2. Using a peptide affinity pull-down approach followed by quantitative mass spectrometry, we identified Copine III as a novel interaction partner of ErbB2-pTyr1248. Copine III belongs to a family of Ca2+-dependent phospholipid binding proteins that is conserved from plants to humans. All copines carry two C2 domains followed by an A domain, similar to the von Willebrand A domain of integrins, in their C-terminus. Although Copine III is ubiquitously expressed, to date it has not been assigned a function downstream of ErbB2. In this study we first analyzed the biochemical properties of Copine III and its interaction with ErbB2. We show that Copine III is a cytoplasmic protein that localizes to the nucleus and the plasma membrane in a Ca2+-dependent manner and upon stimulation of the cells with the ErbB ligand heregulin (HRG). We used FRET acceptor photobleaching to show that Copine III and ErbB2 not only co-localize in HRG-stimulated breast cancer cells, but also interact at the plasma membrane. This co-localization is blocked when the cells are treated with the ErbB2 inhibitor AEE788, implying that Copine III only interacts with phosphorylated active ErbB2. The second goal of my studies was to place Copine III within a signaling pathway downstream of ErbB2. For this, we again used SILAC together with quantitative mass spectrometry and identified the scaffolding protein RACK1 as a binding partner of Copine III. We were able to show that Copine III, RACK1 and the adaptor molecule Shc form a complex with ErbB2 in HRG-stimulated cells. RACK1 has been implicated in focal-adhesion mediated cell migration and here we demonstrate that Copine III localizes to focal adhesions and is required for ErbB2-dependent cell migration. Moreover, knock-down of Copine III affects Src kinase activity and the subsequent phosphorylation of focal adhesion kinase, resulting in the observed defects in cellular migration. Thus Copine III is an important effector molecule in ErbB2-mediated cell migration. Finally, we analyzed Copine III expression in the broader context of cancer, looking at carcinomas of the breast, prostate and ovary. In a set of 49 breast cancer tumor samples, 10 of the 11 cases with ERBB2 amplification display elevated levels of Copine III. This connected well with the protein expression levels of Copine III in a panel of breast cancer cell lines that also correlated with ErbB2 amplification. In published ovarian and prostate transcriptome studies, Copine III mRNA levels are upregulated in cancer as compared to normal tissue. Based on these findings, we performed immunohistochemistry (IHC) stainings of Copine III on breast, prostate and ovarian tissue microarrays. While some Copine III staining was evident in normal breast, normal prostate and ovarian tissues have very low levels of Copine-III. Strikingly, tumors of all three types showed higher Copine III levels. To summarize, we present Copine III here for the first time as an interaction partner of the ErbB2 receptor. Copine III interacts with ErbB2 in a Ca2+- and HRG-dependent manner and is required for tumor cell migration. Furthermore, Copine III levels were found to be upregulated in tissue microarrays of breast, ovarian and prostate tumor tissue as compared to normal tissue. Together, these findings imply a biological function for Copine III in cancer progression and suggest that further studies into the functions of Copine III are merited

Single magnetic adsorbates on s-wave superconductors

In superconductors, magnetic impurities induce a pair-breaking potential for Cooper pairs, which locally affects the Bogoliubov quasiparticles and gives rise to Yu-Shiba-Rusinov (YSR or Shiba, in short) bound states in the density of states (DoS). These states carry information on the magnetic coupling strength of the impurity with the superconductor, which determines the many-body ground state properties of the system. Recently, the interest in Shiba physics was boosted by the prediction of topological superconductivity and Majorana modes in magnetically coupled chains and arrays of Shiba impurities. Here, we review the physical insights obtained by scanning tunneling microscopy into single magnetic adsorbates on the $s$-wave superconductor lead (Pb). We explore the tunneling processes into Shiba states, show how magnetic anisotropy affects many-body excitations, and determine the crossing of the many-body groundstate through a quantum phase transition. Finally, we discuss the coupling of impurities into dimers and chains and their relation to Majorana physics.Comment: 18 pages, 17 figures, revie

Tuning the magnetic anisotropy of single molecules

The magnetism of single atoms and molecules is governed by the atomic scale environment. In general, the reduced symmetry of the surrounding splits the $d$ states and aligns the magnetic moment along certain favorable directions. Here, we show that we can reversibly modify the magnetocrystalline anisotropy by manipulating the environment of single iron(II) porphyrin molecules adsorbed on Pb(111) with the tip of a scanning tunneling microscope. When we decrease the tip--molecule distance, we first observe a small increase followed by an exponential decrease of the axial anisotropy on the molecules. This is in contrast to the monotonous increase observed earlier for the same molecule with an additional axial Cl ligand. We ascribe the changes in the anisotropy of both species to a deformation of the molecules in the presence of the attractive force of the tip, which leads to a change in the $d$ level alignment. These experiments demonstrate the feasibility of a precise tuning of the magnetic anisotropy of an individual molecule by mechanical control.Comment: 16 pages, 5 figures; online at Nano Letters (2015

Visualizing intramolecular distortions as the origin of transverse magnetic anisotropy

The magnetic properties of metal–organic complexes are strongly influenced by conformational changes in the ligand. The flexibility of Fe-tetra-pyridyl-porphyrin molecules leads to different adsorption configurations on a Au(111) surface. By combining low-temperature scanning tunneling spectroscopy and atomic force microscopy, we resolve a correlation of the molecular configuration with different spin states and magnitudes of magnetic anisotropy. When the macrocycle exhibits a laterally undistorted saddle shape, the molecules lie in a S = 1 state with axial anisotropy arising from a square-planar ligand field. If the symmetry in the molecular ligand field is reduced by a lateral distortion of the molecule, we find a finite contribution of transverse anisotropy. Some of the distorted molecules lie in a S = 2 state, again exhibiting substantial transverse anisotropy

Magnetic anisotropy in Shiba bound states across a quantum phase transition

The exchange coupling between magnetic adsorbates and a superconducting substrate leads to Shiba states inside the superconducting energy gap and a Kondo resonance outside the gap. The exchange coupling strength determines whether the quantum many-body ground state is a Kondo singlet or a singlet of the paired superconducting quasiparticles. Here, we use scanning tunneling spectroscopy to identify the different quantum ground states of Manganese phthalocyanine on Pb(111). We observe Shiba states, which are split into triplets by magnetocrystalline anisotropy. Their characteristic spectral weight yields an unambiguous proof of the nature of the quantum ground state.Comment: 6 pages, 4 figure

End states and subgap structure in proximity-coupled chains of magnetic adatoms

A recent experiment [Nadj-Perge et al., Science 346, 602 (2014)] provides evidence for Majorana zero modes in iron (Fe) chains on the superconducting Pb(110) surface. Here, we study this system by scanning tunneling microscopy using superconducting tips. This high-resolution technique resolves a rich subgap structure, including zero-energy excitations in some chains. We compare the symmetry properties of the data under voltage reversal against theoretical expectations and provide evidence that the putative Majorana signature overlaps with a previously unresolved low-energy resonance. Interpreting the data within a Majorana framework suggests that the topological gap is significantly smaller than previously believed. Aided by model calculations, we also analyze higher-energy features of the subgap spectrum and their relation to high-bias peaks which we associate with the Fe d-bands.Comment: 5+5 pages, 5+6 figure