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Milena Hering

Publications and source records attributed to Milena Hering.

16 recordsLinked to original sources
arXiv2025

F-Splittings of seminormal monoid algebras

We compute a number of invariants of singularities defined via the Frobenius morphism for seminormal affine toric varieties over fields of characteristic p > 0. Our main technical tool is a combinatorial description of the potential splittings of iterates of Frobenius for seminormal monoid algebras. This allows us to give an easy formula for the F-splitting ratio of such rings as well as to compute the ideals stable under the Cartier algebra, including the test ideal.

math.AC↗
arXiv2024

On the instability of syzygy bundles on toric surfaces

We show that for every toric surface apart from the projective plane and a product of two projective lines and every ample line bundle there exists a polarisation such that the syzygy bundle associated to sufficiently high powers of the line bundle is not slope stable.

math.AG↗
arXiv2021

Toric and tropical Bertini theorems in positive characteristic

We generalize the toric Bertini theorem of Fuchs, Mantova, and Zannier to positive characteristic. A key part of the proof is a new algebraically closed field containing the field \kk(t_1,\dots,t_d) of rational functions over an algebraically closed field \kk of prime characteristic. As a corollary, we extend the tropical Bertini theorem of Maclagan and Yu to arbitrary characteristic, which removes the characteristic dependence from the d-connectivity result for tropical varieties from that paper.

math.AG↗
arXiv2019

Stability of tangent bundles on smooth toric Picard-rank-2 varieties and surfaces

We give a combinatorial criterion for the tangent bundle on a smooth toric variety to be stable with respect to a given polarisation in terms of the corresponding lattice polytope. Furthermore, we show that for a smooth toric surface and a smooth toric variety of Picard rank 2, there exists an ample line bundle with respect to which the tangent bundle is stable if and only if it is an iterated blow-up of projective space.

math.AG↗
arXiv2016

Diagonal splittings of toric varieties and unimodularity

We use a polyhedral criterion for the existence of diagonal splittings to investigate which toric varieties X are diagonally split. Our results are stated in terms of the vector configuration given by primitive generators of the 1-dimensional cones in the fan defining X. We show, in particular, that X is diagonally split at all q if and only if this configuration is unimodular, and X is not diagonally split at any q if this configuration is not 2-regular. We also study implications for the possibilities for the set of q at which a toric variety X is diagonally split.

math.AG↗
arXiv2016

Toric degenerations of Gr(2,n) and Gr(3,6) via plabic graphs

We establish an explicit bijection between the toric degenerations of the Grassmannian $\textrm{Gr}(2,n)$ arising from maximal cones in tropical Grassmannians and the ones coming from plabic graphs corresponding to $\textrm{Gr}(2,n)$. We show that a similar statement does not hold for $\textrm{Gr}(3,6)$.

math.CO↗
arXiv2014

The ring of evenly weighted points on the line

Let $M_w = (\Pj^1)^n \q \mathrm{SL}_2$ denote the geometric invariant theory quotient of $(\Pj^1)^n$ by the diagonal action of $\mathrm{SL}_2$ using the line bundle $\mathcal{O}(w_1,w_2,...,w_n)$ on $(\Pj^1)^n$. Let $R_w$ be the coordinate ring of $M_w$. We give a closed formula for the Hilbert function of $R_w$, which allows us to compute the degree of $M_w$. The graded parts of $R_w$ are certain Kostka numbers, so this Hilbert function computes stretched Kostka numbers. If all the weights $w_i$ are even, we find a presentation of $R_w$ so that the ideal $I$ of this presentation has a quadratic Gröbner basis. In particular, $R_w$ is Koszul. We obtain this result by studying the homogeneous coordinate ring of a projective toric variety arising as a degeneration of $M_w$.

math.AG↗
arXiv2013

An algebraic characterization of injectivity in phase retrieval

A complex frame is a collection of vectors that span $\mathbb{C}^M$ and define measurements, called intensity measurements, on vectors in $\mathbb{C}^M$. In purely mathematical terms, the problem of phase retrieval is to recover a complex vector from its intensity measurements, namely the modulus of its inner product with these frame vectors. We show that any vector is uniquely determined (up to a global phase factor) from $4M-4$ generic measurements. To prove this, we identify the set of frames defining non-injective measurements with the projection of a real variety and bound its dimension.

math.FA↗
arXiv2013

Few smooth d-polytopes with n lattice points

We prove that, for fixed n there exist only finitely many embeddings of Q-factorial toric varieties X into P^n that are induced by a complete linear system. The proof is based on a combinatorial result that for fixed nonnegative integers d and n, there are only finitely many smooth d-polytopes with n lattice points. We also enumerate all smooth 3-polytopes with at most 12 lattice points. In fact, it is sufficient to bound the singularities and the number of lattice points on edges to prove finiteness.

math.AG↗
arXiv2011

The T-graph of a multigraded Hilbert scheme

The T-graph of a multigraded Hilbert scheme records the zero and one-dimensional orbits of the T = (K^*)^n action on the Hilbert scheme induced from the T-action on A^n. It has vertices the T-fixed points, and edges the one-dimensional T-orbits. We give a combinatorial necessary condition for the existence of an edge between two vertices in this graph. For the Hilbert scheme of points in the plane, we give an explicit combinatorial description of the equations defining the scheme parameterizing all one-dimensional torus orbits whose closures contain two given monomial ideals. For this Hilbert scheme we show that the T-graph depends on the ground field, resolving a question of Altmann and Sturmfels.

math.AG↗
arXiv2011

Cox rings and pseudoeffective cones of projectivized toric vector bundles

We study projectivizations of a special class of toric vector bundles that includes cotangent bundles, whose associated Klyachko filtrations are particularly simple. For these projectivized bundles, we give generators for the cone of effective divisors and a presentation of the Cox ring as a polynomial algebra over the Cox ring of a blowup of a projective space along a sequence of linear subspaces. As applications, we show that the projectivized cotangent bundles of some toric varieties are not Mori dream spaces and give examples of projectivized toric vector bundles whose Cox rings are isomorphic to that of M_{0,n}.

math.AG↗
arXiv2009

Positivity for toric vector bundles

We show that an equivariant vector bundle on a complete toric variety is nef or ample if and only if its restriction to every invariant curve is nef or ample, respectively. Furthermore, we show that nef toric vector bundles have a nonvanishing global section at every point, and deduce that the underlying vector bundle is trivial if and only if its restriction to every invariant curve is trivial. We apply our methods and results to study, in particular, the vector bundles M_L that arise as the kernel of the evaluation map on sections of L, when L is an ample line bundle. We give examples of twists of such bundles that are ample but not globally generated.

math.AG↗
arXiv2006

Syzygies, multigraded regularity and toric varieties

Using multigraded Castelnuovo-Mumford regularity, we study the equations defining a projective embedding of a variety X. Given globally generated line bundles B_1, ..., B_k on X and integers m_1, ..., m_k, consider the line bundle L := B_1^m_1 \otimes ... \otimes B_k^m_k. We give conditions on the m_i which guarantee that the ideal of X in P(H^0(X,L)) is generated by quadrics and the first p syzygies are linear. This yields new results on the syzygies of toric varieties and the normality of polytopes.

math.AG↗
arXiv2006

Asymptotic cohomological functions of toric divisors

We study functions on the class group of a toric variety measuring the rates of growth of the cohomology groups of multiples of divisors. We show that these functions are piecewise polynomial with respect to finite polyhedral chamber decompositions. As applications, we express the self-intersection number of a T-Cartier divisor as a linear combination of the volumes of the bounded regions in the corresponding hyperplane arrangement and prove an asymptotic converse to Serre vanishing.

math.AG↗
arXiv2004

Syzygies, regularity and toric varieties

Let A be an ample line bundle on a projective toric variety X of dimension n. We show that if l>=n-1+p, then A^l satisfies the property N_p. Applying similar methods, we obtain a combinatorial theorem: For a given lattice polytope P we give a criterion for an integer m to guarantee that mP is normal.

math.AG↗