arXiv subjects
Publications and source records attributed to Philip Engel.
We construct a modular compactification via stable slc pairs for the moduli spaces of K3 surfaces with a nonsymplectic group of automorphisms under the assumption that some combination of the fixed loci of automorphisms defines an effective big divisor, and prove that it is semitoroidal.
A cusp singularity is a surface singularity whose minimal resolution is a cycle of smooth rational curves meeting transversely. Cusp singularities come in naturally dual pairs. Looijenga proved in 1981 that if a cusp singularity is smoothable, the minimal resolution of the dual cusp is the anticanonical divisor of some smooth rational surface. In 1983, the second author and Miranda gave a criterion for smoothability of a cusp singularity, in terms of the existence of a K-trivial semistable model for the central fiber of such a smoothing. We study these "Type III degenerations" of rational surfaces with an anticanonical divisor--their deformations, birational geometry, and monodromy. Looijenga's original paper also gave a description of the rational double point configurations to which a cusp singularity deforms, but only in the case where the resolution of the dual cusp has cycle length 5 or less. We generalize this classification to an arbitrary cusp singularity, giving an explicit construction of a semistable simultaneous resolution of such an adjacency. The main tools of the proof are (1) formulas for the monodromy of a Type III degeneration, (2) a construction via surgeries on integral-affine surfaces of a degeneration with prescribed monodromy, (3) surjectivity of the period map for Type III central fibers, and (4) a theorem of Shepherd-Barron producing the simultaneous contraction to the adjacency of the cusp singularity.
The flex divisor of a primitively polarized K3 surface $(X,L)$ of degree $L^2=2d$ is, generically, the locus of all points $x\in X$ for which there exists a pencil $V\subset |L|$ whose base locus is $\{x\}$. We show that the flex divisor lies in the linear system $|n_dL|$ where $n_d=(2d+1)C(d)^2$ and $C(d)$ is the Catalan number. We also show that there is a well-defined notion of flex divisor over the whole moduli space $F_{2d}$ of polarized K3 surfaces.
We describe two geometrically meaningful compactifications of the moduli space of elliptic K3 surfaces via stable slc pairs, for two different choices of a polarizing divisor, and show that their normalizations are two different toroidal compactifications of the moduli space, one for the ramification divisor and another for the rational curve divisor. In the course of the proof, we further develop the theory of integral affine spheres with 24 singularities. We also construct moduli of rational (generalized) elliptic stable slc surfaces of types ${\bf A_n}$ ($n\ge1$), ${\bf C_n}$ ($n\ge0$) and ${\bf E_n}$ ($n\ge0$).
An elliptic orbifold is the quotient of an elliptic curve by a finite group. Eskin and Okounkov proved that generating functions for the number of branched covers of an elliptic curve with specified ramification are quasimodular forms for the full modular group $SL_2(\mathbb{Z})$. They later generalized this theorem to the enumeration of branched covers of a pillowcase, i.e. the quotient of an elliptic curve by the elliptic involution, proving quasi-modularity for $Γ_1(2)$. We generalize their work to the quotient of an elliptic curve by cyclic groups of orders $N=3$, $4$, $6$, proving quasi-modularity for level $Γ_1(N)$. One corollary is that certain generating functions of hexagon, square, and triangle tilings of compact surfaces are quasi-modular. These tilings enumerate lattice points in moduli spaces of flat surfaces. We analyze the asymptotic behavior as the number of tiles goes to infinity, theoretically giving an algorithm to compute the Masur-Veech volumes of moduli spaces of cubic, quartic, and sextic differentials. We also deduce that the volume is polynomial in $π$.
An elliptic orbifold is the quotient of an elliptic curve by a finite group. In 2001, Eskin and Okounkov proved that generating functions for the number of branched covers of an elliptic curve with specified ramification are quasimodular forms for $SL_2(\mathbb{Z}).$ In 2006, they generalized this theorem to the enumeration of branched covers of the quotient of an elliptic curve by $\pm 1$, proving quasi-modularity for $Γ_1(2)$. In 2017, the author generalized their work to the quotient of an elliptic curve by $\langle ζ_N\rangle$ for $N=3, 4, 6$, proving quasimodularity for $Γ_1(N)$. In these works, both Eskin-Okounkov and the author had to assume that there was at least one orbifold point of order $N$ over which there was no ramification. Here we remove that assumption, with the caveat that the generating functions are only quasimodular for $Γ(N)$. We deduce the following corollary: Let $h_6(\vecκ,q)$ be the generating function whose $q^n$ coefficient is the number of surface triangulations with $2n$ triangles, such that the set of non-zero curvatures is $κ_i$. Here the curvature of a vertex is six minus its valence. Then under the substitution $q=e^{2πi τ}$, the function $h_6(\vecκ,q)$ is a quasimodular form for $Γ_1(6)$ with weight bounded in terms of $\vecκ$. This statement in turn implies that the Masur-Veech volume of any stratum of sextic differentials is polynomial in $π$.
A cusp singularity is an elliptic surface singularity whose minimal resolution is a cycle of smooth rational curves meeting transversely. Cusp singularities come in naturally dual pairs. In 1981, Looijenga proved that whenever a cusp singularity is smoothable, the minimal resolution of the dual cusp is an anticanonical divisor of some smooth rational surface. He conjectured the converse. Recent work of Gross, Hacking, and Keel has proven Looijenga's conjecture using methods from mirror symmetry. This paper provides an alternative proof of Looijenga's conjecture based on a combinatorial criterion for smoothability given by Friedman and Miranda in 1983.
A tiling of the sphere by triangles, squares, or hexagons is convex if every vertex has at most 6, 4, or 3 polygons adjacent to it, respectively. Assigning an appropriate weight to any tiling, our main result is explicit formulas for the weighted number of convex tilings with a given number of tiles. To prove these formulas, we build on work of Thurston, who showed that the convex triangulations correspond to orbits of vectors of positive norm in a Hermitian lattice $Λ\subset \mathbb{C}^{1,9}$. First, we extend this result to convex square- and hexagon-tilings. Then, we explicitly compute the relevant lattice $Λ$. Next, we integrate the Siegel theta function for $Λ$ to produce a modular form whose Fourier coefficients encode the weighted number of tilings. Finally, we determine the formulas using finite-dimensionality of spaces of modular forms.
We report the realization of a device based on a single Nitrogen-vacancy (NV) center in diamond coupled to a fiber-cavity for use as single photon source (SPS). The device consists of two concave mirrors each directly fabricated on the facets of two optical fibers and a preselected nanodiamond containing a single NV center deposited onto one of these mirrors. Both, cavity in- and output are directly fiber-coupled and the emission wavelength is easily tunable by variation of the separation of the two mirrors with a piezo-electric crystal. By coupling to the cavity we achieve an increase of the spectral photon rate density by two orders of magnitude compared to free-space emission of the NV center. With this work we establish a simple all-fiber based SPS with promising prospects for the integration into photonic quantum networks.
Their intrinsic properties render single quantum systems as ideal tools for quantum enhanced sensing and microscopy. As an additional benefit, their size is typically on an atomic scale which enables sensing with very high spatial resolution. Here, we report on utilizing a single nitrogen vacancy center in nanodiamond for performing three-dimensional scanning-probe fluorescence lifetime imaging microscopy. By measuring changes of the single emitter's lifetime information on the local density of optical states is acquired at the nanoscale. This technique to gather information on the local density of optical states is important for the understanding of fundamental quantum optical processes as well as for the engineering of novel photonic and plasmonic devices.