19 citations · 66 across the 16 of their papers we have counts for
20 papers
Complex structures on
Philip Engel
A recent manuscript produced by Claude under the direction of Levent Alpöge constructs a 1-parameter family of compact complex threefolds diffeomorphic to . Our goal is to des…
Polync varieties and multiparameter Kulikov models
Philip Engel
We study "polync varieties", whose singularities are locally products of normal crossing (nc) singularities. We introduce the notion of d-semistability of such varieties, and gener…
On lattice-polarized K3 surfaces
Valery Alexeev, Philip Engel
We propose modifications to the commonly used definitions of lattice-polarized and lattice-quasipolarized smooth K3 surfaces, collecting various versions of the definition, and det…
Shafarevich's conjecture for families of hypersurfaces over function fields
Philip Engel, Alice Lin, Salim Tayou
Given a smooth quasi-projective complex algebraic variety , we prove that there are only finitely many Hodge-generic non-isotrivial families of smooth projective hyper…
Compact moduli of Enriques surfaces with a numerical polarization of degree 2
Valery Alexeev, Philip Engel, D. Zack Garza +1
We describe a geometric, stable pair compactification of the moduli space of Enriques surfaces with a numerical polarization of degree 2, and identify it with a semitoroidal compac…
Periods of elliptic surfaces with
Philip Engel, François Greer, Abigail Ward
We prove that the period mapping is dominant for elliptic surfaces over an elliptic curve with 12 nodal fibers, and that its degree is larger than 1.
On the non-abelian Hodge locus I
Philip Engel, Salim Tayou
We partially resolve conjectures of Deligne and Simpson concerning -local systems on quasi-projective varieties that underlie a polarized variation of Hodge structure.…
Mixed mock modularity of special divisors
Philip Engel, François Greer, Salim Tayou
We prove that the generating series of special divisors in toroidal compactifications of orthogonal Shimura varieties is a mixed mock modular form. More precisely, we find an expli…
The Manin-Mumford conjecture in genus 2 and rational curves on K3 surfaces
Philip Engel, Raju Krishnamoorthy, Daniel Litt
Let be a simple abelian surface over an algebraically closed field . Let be the set of torsion points of such that there exists a genus curve …
Compactifications of moduli spaces of K3 surfaces with a nonsymplectic involution
Valery Alexeev, Philip Engel
There are moduli spaces of K3 surfaces with a nonsymplectic involution. We give detailed descriptions of Kulikov models for one-parameter degenerations in . In the…
Compact moduli of K3 surfaces with a nonsymplectic automorphism
Valery Alexeev, Philip Engel, Changho Han
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 combina…
The flex divisor of a K3 surface
Valery Alexeev, Philip Engel
The flex divisor of a primitively polarized K3 surface of degree is, generically, the locus of all points for which there exists a pencil w…
Compact moduli of K3 surfaces
Valery Alexeev, Philip Engel
We construct geometric compactifications of the moduli space of polarized K3 surfaces, in any degree . Our construction is via KSBA theory, by considering canonical ch…
Compactifications of moduli of elliptic K3 surfaces: stable pair and toroidal
Valery Alexeev, Adrian Brunyate, Philip Engel
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…
Stable pair compactification of moduli of K3 surfaces of degree 2
Valery Alexeev, Philip Engel, Alan Thompson
We prove that the universal family of polarized K3 surfaces of degree 2 can be extended to a flat family of stable slc pairs over the toroidal compactification associated…
Hurwitz Theory of Elliptic Orbifolds, II
Philip Engel
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 el…
Hurwitz Theory of Elliptic Orbifolds, I
Philip Engel
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 cu…
The number of convex tilings of the sphere by triangles, squares, or hexagons
Philip Engel, Peter Smillie
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…
Smoothings and Rational Double Point Adjacencies for Cusp Singularities
Philip Engel, Robert Friedman
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. Loo…
Looijenga's Conjecture via Integral-affine Geometry
Philip Engel
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…