arXiv subjects

Takumi Murayama

Publications and source records attributed to Takumi Murayama.

23 records · Page 2Linked to original sources

arXiv2019

Seshadri constants for vector bundles

We introduce Seshadri constants for line bundles in a relative setting. They generalize the classical Seshadri constants of line bundles on projective varieties and their extension to vector bundles studied by Beltrametti-Schneider-Sommese and Hacon. There are similarities to the classical theory. In particular, we give a Seshadri-type ampleness criterion, and we relate Seshadri constants to jet separation and to asymptotic base loci. We give three applications of our new version of Seshadri constants. First, a celebrated result of Mori can be restated as saying that any Fano manifold whose tangent bundle has positive Seshadri constant at a point is isomorphic to a projective space. We conjecture that the Fano condition can be removed. Among other results in this direction, we prove the conjecture for surfaces. Second, we restate a classical conjecture on the nef cone of self-products of curves in terms of semistability of higher conormal sheaves, which we use to identify new nef classes on self-products of curves. Third, we prove that our Seshadri constants can be used to control separation of jets for direct images of pluricanonical bundles, in the spirit of a relative Fujita-type conjecture of Popa and Schnell.

math.AG↗

arXiv2018

Effective generation and twisted weak positivity of direct images

In this paper, we study pushforwards of log pluricanonical bundles on projective log canonical pairs $(Y,Δ)$ over the complex numbers, partially answering a Fujita-type conjecture due to Popa and Schnell in the log canonical setting. We show two effective global generation results. First, when $Y$ surjects onto a projective variety, we show a quadratic bound for generic generation for twists by big and nef line bundles. Second, when $Y$ is fibered over a smooth projective variety, we show a linear bound for twists by ample line bundles. These results additionally give effective non-vanishing statements. We also prove an effective weak positivity statement for log pluricanonical bundles in this setting, which may be of independent interest. In each context we indicate over which loci positivity holds. Finally, using the description of such loci, we show an effective vanishing theorem for pushforwards of certain log-sheaves under smooth morphisms.

math.AG↗

arXiv2018

Frobenius-Seshadri constants and characterizations of projective space

We introduce higher-order variants of the Frobenius-Seshadri constant due to Mustaţă and Schwede, which are defined for ample line bundles in positive characteristic. These constants are used to show that Demailly's criterion for separation of higher-order jets by adjoint bundles also holds in positive characteristic. As an application, we give a characterization of projective space using Seshadri constants in positive characteristic, which was proved in characteristic zero by Bauer and Szemberg. We also discuss connections with other characterizations of projective space.

math.AG↗

arXiv2016

Dilated Floor Functions That Commute

We determine all pairs of real numbers $(α, β)$ such that the dilated floor functions $\lfloor αx\rfloor$ and $\lfloor βx\rfloor$ commute under composition, i.e., such that $\lfloor α\lfloor βx\rfloor\rfloor = \lfloor β\lfloor αx\rfloor\rfloor$ holds for all real $x$.

math.NT↗

arXiv2013

Robust Graph Ideals

Let I be a toric ideal. We say I is robust if its universal Groebner basis is a minimal generating set. We show that any robust toric ideal arising from a graph G is also minimally generated by its Graver basis. We then completely characterize all graphs which give rise to robust ideals. Our characterization shows that robustness can be determined solely in terms of graph-theoretic conditions on the set of circuits of G.

math.CO↗