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Creators/Authors contains: "ENGEL, PHILIP"

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  1. ; ; (, International Mathematics Research Notices)
    Given a smooth quasi-projective complex algebraic variety $$\mathcal{S}$$, we prove that there are only finitely many Hodge-generic non-isotrivial families of smooth projective hypersurfaces over $$\mathcal{S}$$ of degree $$d$$ in $$\mathbb{P}_{\mathbb C}^{n+1}$$. We prove that the finiteness is uniform in $$\mathcal{S}$$ and give examples where the result is sharp. We also prove similar results for certain complete intersections in $$\mathbb{P}_{\mathbb C}^{n+1}$$ of higher codimension and more generally for algebraic varieties whose moduli space admits a period map that satisfies the infinitesimal Torelli theorem. 
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    Free, publicly-accessible full text available August 1, 2026
  2. ; ; (, Forum of Mathematics, Sigma)
    Abstract 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$$1$$. This settles the final case of infinitesimal Torelli for a generic elliptic surface. 
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    Free, publicly-accessible full text available November 26, 2025
  3. ; ; ; (, Nagoya Mathematical Journal)
    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 compactification of the period space. 
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    Free, publicly-accessible full text available February 26, 2026
  4. ; ; (, Duke Mathematical Journal)
    A fullerene, or buckyball, is a trivalent graph on the sphere with only pentagonal and hexagonal faces. Building on ideas of Thurston, we use modular forms to give an exact formula for the number of oriented fullerenes with a given number of vertices. 
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    Free, publicly-accessible full text available February 15, 2026
  5. ; (, Annals of Mathematics)
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  6. ; ; (, Journal für die reine und angewandte Mathematik (Crelles Journal))
    We prove that the universal family of polarized K3 surfaces of degree 2 can be extended to a flat family of stable KSBA pairs over the toroidal compactification associated to the Coxeter fan. One-parameter degenerations of K3 surfaces in this family are described by integral-affine structures on a sphere with 24 singularities. 
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    Full Text Available 
  7. ; (, International Mathematics Research Notices)
    Abstract The flex divisor$$R_{\textrm flex}$$ of a primitively polarized K3 surface $(X,L)$ is, generically, the set of all points $$x\in X$$ for which there exists a pencil $$V\subset |L|$$ whose base locus is $$\{x\}$$. We show that if $L^2=2d$ then $$R_{\textrm flex}\in |n_dL|$$ with $$ \begin{align*} &n_d= \frac{(2d)!(2d+1)!}{d!^2(d+1)!^2} =(2d+1)C(d)^2,\end{align*}$$where $C(d)$ 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. 
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  8. ; ; (, Geometry & Topology)
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