arXiv subjects

Mara Ungureanu

Publications and source records attributed to Mara Ungureanu.

6 recordsLinked to original sources

arXiv2026

Tracking the symmetries of $\mathbb Z_3$-orbifold K3s within the Mathieu groups

For $\mathbb Z_3$-orbifold limits of K3, we provide a counterpart to the extensive studies by Nikulin and others of the geometry and symmetries of classical Kummer surfaces. In particular, we determine the group of holomorphic symplectic automorphisms of $\mathbb Z_3$-orbifold limits of K3. We moreover track this group within two of the Mathieu groups, which involves a variation of Kondo's lattice techniques that Taormina and Wendland introduced earlier in their study of the symmetries of Kummer surfaces and the genesis of their symmetry surfing programme. Specifically, we realise the finite group of symplectic automorphisms of this class of K3 surfaces as a subgroup of the sporadic groups Mathieu 12 and Mathieu 24 in terms of permutations of 12, resp. 24 elements. As a proof of concept, we construct an embedding that yields the largest Mathieu group when the symmetry group of $\mathbb Z_3$-orbifold K3s is combined with all symmetries of Kummer surfaces.

math.AG↗

arXiv2020

Universal polynomials for counts of secant planes to projective curves

In this article we provide another method for obtaining explicit formulas yielding counts of secant planes to a projective curve. We formulate the problem in terms of Segre classes of suitable bundles over the symmetric product of the curve and take advantage of the result of Ellingsrud, Göttsche, and Lehn concerning the structure of integrals of polynomials in Chern classes of tautological bundles over Hilbert schemes of points. We use this to set up a recursion starting from the easy to compute case of the projective line.

math.AG↗

arXiv2020

Geometry of intersections of some secant varieties to algebraic curves

For a smooth projective curve, the cycles of subordinate or, more generally, secant divisors to a given linear series are among some of the most studied objects in classical enumerative geometry. We consider the intersection of two such cycles corresponding to secant divisors of two different linear series on the same curve and investigate the validity of the enumerative formulas counting the number of divisors in the intersection. We study some interesting cases, with unexpected transversality properties, and establish a general method to verify when this intersection is empty.

math.AG↗

arXiv2019

Refined de Jonquières divisors and secant varieties on algebraic curves

The aim of this paper is to provide another perspective on secant varieties on algebraic curves by reformulating the problem in terms of refined de Jonquières divisors, that is divisors on the curve with prescribed multiplicities and dimensions of their spaces of global sections. We are able to both recover some already known results and to obtain some new statements concerning the dimension theory of secant varieties. We do this via the study of the dimension theory of refined de Jonquières divisors in some relevant cases and degeneration arguments.

math.AG↗

arXiv2019

Dimension theory and degenerations of de Jonquières divisors

This paper aims at settling the issue of the validity of the de Jonquières formulas. We consider the space of divisors with prescribed multiplicity, or de Jonquières divisors, contained in a linear series on a smooth projective curve. Assuming zero expected dimension of this space, the de Jonquières formulas compute the virtual number of de Jonquières divisors. Using degenerations to nodal curves we show that for a general curve equipped with a general complete linear series, the space is of expected dimension, which shows that the counts are in fact true. This implies that in the case of negative expected dimension a general linear series on a general curve does not admit de Jonquières divisors of the expected type.

math.AG↗

arXiv2017

Homogeneous M2 duals

Motivated by the search for new gravity duals to M2 branes with $N>4$ supersymmetry --- equivalently, M-theory backgrounds with Killing superalgebra $\mathfrak{osp}(N|4)$ for $N>4$ --- we classify homogeneous M-theory backgrounds with symmetry Lie algebra $\mathfrak{so}(n) \oplus \mathfrak{so}(3,2)$ for $n=5,6,7$. We find that there are no new backgrounds with $n=6,7$ but we do find a number of new (to us) backgrounds with $n=5$. All backgrounds are metrically products of the form $\operatorname{AdS}_4 \times P^7$, with $P$ riemannian and homogeneous under the action of $\operatorname{SO}(5)$, or $S^4 \times Q^7$ with $Q$ lorentzian and homogeneous under the action of $\operatorname{SO}(3,2)$. At least one of the new backgrounds is supersymmetric (albeit with only $N=2$) and we show that it can be constructed from a supersymmetric Freund--Rubin background via a Wick rotation. Two of the new backgrounds have only been approximated numerically. (The second version of this paper includes an appendix by Alexander~S.~Haupt, closing a gap in our original analysis.)

hep-th↗