Generalized Borcea-Voisin Construction

C. Voisin and C. Borcea have constructed mirror pairs of families of Calabi-Yau threefolds by taking the quotient of the product of an elliptic curve with a K3 surface endowed with a non-symplectic involution. In this paper, we generalize the construction of Borcea and Voisin to any prime order and build three and four dimensional Calabi-Yau orbifolds. We classify the topological types that are obtained and show that, in dimension 4, orbifolds built with an involution admit a crepant resolution and come in topological mirror pairs. We show that for odd primes, there are generically no minimal resolutions and the mirror pairing is lost.Comment: 15 pages, 2 figures. v2: typos corrected & references adde

Brill–Noether general K3 surfaces with the maximal number of elliptic pencils of minimal degree

We explicitly construct Brill–Noether general K3 surfaces of genus 4, 6 and 8 having the maximal number of elliptic pencils of degrees 3, 4 and 5, respectively, and study their moduli spaces and moduli maps to the moduli space of curves. As an application we prove the existence of Brill–Noether general K3 surfaces of genus 4 and 6 without stable Lazarsfeld–Mukai bundles of minimal c2.publishedVersio

Geometric invariant theory of syzygies, with applications to moduli spaces

We define syzygy points of projective schemes, and introduce a program of studying their GIT stability. Then we describe two cases where we have managed to make some progress in this program, that of polarized K3 surfaces of odd genus, and of genus six canonical curves. Applications of our results include effectivity statements for divisor classes on the moduli space of odd genus K3 surfaces, and a new construction in the Hassett-Keel program for the moduli space of genus six curves.Comment: v1: 23 pages, submitted to the Proceedings of the Abel Symposium 2017, v2: final version, corrects a sign error and resulting divisor class calculations on the moduli space of K3 surfaces in Section 5, other minor changes, In: Christophersen J., Ranestad K. (eds) Geometry of Moduli. Abelsymposium 2017. Abel Symposia, vol 14. Springer, Cha

Computing Linear Matrix Representations of Helton-Vinnikov Curves

Helton and Vinnikov showed that every rigidly convex curve in the real plane bounds a spectrahedron. This leads to the computational problem of explicitly producing a symmetric (positive definite) linear determinantal representation for a given curve. We study three approaches to this problem: an algebraic approach via solving polynomial equations, a geometric approach via contact curves, and an analytic approach via theta functions. These are explained, compared, and tested experimentally for low degree instances.Comment: 19 pages, 3 figures, minor revisions; Mathematical Methods in Systems, Optimization and Control, Birkhauser, Base

K3 surfaces and log del Pezzo surfaces of index three

We use classification of non-symplectic automorphisms of K3 surfaces to obtain a partial classification of log del Pezzo surfaces of index three. We can classify those with "Multiple Smooth Divisor Property", whose definition we will give. Our methods include the definition of right resolutions of quotient singularities of index three and some analysis of automorphism-stable elliptic fibrations on K3 surfaces. In particular we find several log del Pezzo surfaces of Picard number one with non-toric singularities of index three.Comment: 32 pages, to appear in Manuscripta Mat

From SICs and MUBs to Eddington

This is a survey of some very old knowledge about Mutually Unbiased Bases (MUB) and Symmetric Informationally Complete POVMs (SIC). In prime dimensions the former are closely tied to an elliptic normal curve symmetric under the Heisenberg group, while the latter are believed to be orbits under the Heisenberg group in all dimensions. In dimensions 3 and 4 the SICs are understandable in terms of elliptic curves, but a general statement escapes us. The geometry of the SICs in 3 and 4 dimensions is discussed in some detail.Comment: 12 pages; from the Festschrift for Tony Sudber

Alternating groups and moduli space lifting Invariants

Main Theorem: Spaces of r-branch point 3-cycle covers, degree n or Galois of degree n!/2 have one (resp. two) component(s) if r=n-1 (resp. r\ge n). Improves Fried-Serre on deciding when sphere covers with odd-order branching lift to unramified Spin covers. We produce Hurwitz-Torelli automorphic functions on Hurwitz spaces, and draw Inverse Galois conclusions. Example: Absolute spaces of 3-cycle covers with +1 (resp. -1) lift invariant carry canonical even (resp. odd) theta functions when r is even (resp. odd). For inner spaces the result is independent of r. Another use appears in, http://www.math.uci.edu/~mfried/paplist-mt/twoorbit.html, "Connectedness of families of sphere covers of A_n-Type." This shows the M(odular) T(ower)s for the prime p=2 lying over Hurwitz spaces first studied by, http://www.math.uci.edu/~mfried/othlist-cov/hurwitzLiu-Oss.pdf, Liu and Osserman have 2-cusps. That is sufficient to establish the Main Conjecture: (*) High tower levels are general-type varieties and have no rational points.For infinitely many of those MTs, the tree of cusps contains a subtree -- a spire -- isomorphic to the tree of cusps on a modular curve tower. This makes plausible a version of Serre's O(pen) I(mage) T(heorem) on such MTs. Establishing these modular curve-like properties opens, to MTs, modular curve-like thinking where modular curves have never gone before. A fuller html description of this paper is at http://www.math.uci.edu/~mfried/paplist-cov/hf-can0611591.html .Comment: To appear in the Israel Journal as of 1/5/09; v4 is corrected from proof sheets, but does include some proof simplification in \S

Birational geometry of hypersurfaces in products of projective spaces

We study the birational properties of hypersurfaces in products of projective spaces. In the case of hypersurfaces in Pm x Pn, we describe their nef, movable and e ective cones and determine when they are Mori dream spaces. Using this, we give new simple examples of non-Mori dream spaces and analogues of Mumford's example of a strictly nef line bundle which is not ample.This is the author accepted manuscript. The final version is available from Springer via http://dx.doi.org/10.1007/s00209-015-1415-