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Mihai Fulger

Publications and source records attributed to Mihai Fulger.

17 recordsLinked to original sources
arXiv2025

Fujita-Zariski decompositions on some product threefolds

We use explicit blow-ups and computations of birational Fujita-Zariski decompositions to determine generic infinitesimal Newton-Okounkov bodies for box-product ample polarizations on three classes of spaces: product between a curve and the projective plane, products of three curves, and the product between a curve and a Jacobian surface. In particular we compute the volume of many big but non-nef divisors on blow-ups of these threefolds.

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arXiv2025

Infinitesimal successive minima, partial jets and convex geometry

We introduce two sets of invariants for a line bundle at a point: infinitesimal successive minima and asymptotic partial jet separation. They are inspired by the local analogue of Ambro-Ito, and by the jet-theoretic interpretation of the Seshadri constant respectively. Under mild restrictions the two sets are equal. Moving to convex geometry, we prove that the lengths of the maximal simplex inside the generic infinitesimal Newton-Okounkov body (iNObody) of the line bundle at the point are precisely the successive minima. As application we characterize when this body is simplicial, and give examples when it is not. When the point is very general the convex body has a shape that we call Borel-fixed, a property inspired by generic initial ideals. Borel-fixed convex bodies satisfy simplicial lower bounds and polytopal upper bounds determined by their widths. For the generic iNObody of the line bundle at very general points these widths are again the infinitesimal successive minima.

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arXiv2023

Positivity and base loci for vector bundles revisited

We give equivalent descriptions for the augmented and diminished base loci of vector bundles in characteristic zero. We show that these base loci behave well under pullback, tensor product, and direct sum. Pathological behavior is observed on some nonsplit exact sequences.

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arXiv2021

New constructions of nef classes on self-products of curves

We study the nef cone of self-products of a curve. When the curve is very general of genus $g>2$, we construct a nontrivial class of self-intersection 0 on the boundary of the nef cone. Up to symmetry, this is the only known nontrivial boundary example that exists for all $g > 2$. When the curve is general, we identify nef classes that improve on known examples for arbitrary curves. We also consider self-products of more than two copies of the curve.

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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.

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arXiv2018

Seshadri constants for curve classes

We develop a local positivity theory for movable curves on projective varieties similar to the classical Seshadri constants of nef divisors. We give analogues of the Seshadri ampleness criterion, of a characterization of the augmented base locus of a big and nef divisor, and of the interpretation of Seshadri constants as an asymptotic measure of jet separation. We also study the case of arbitrary codimension.

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arXiv2016

Newton-Okounkov bodies and complexity functions

We show that quite universally the holonomicity of the complexity function of a big divisor on a projective variety does not predict the polyhedrality of the Newton-Okounkov body associated to every flag.

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arXiv2016

Zariski decompositions of numerical cycle classes

We construct a Zariski decomposition for cycle classes of arbitrary codimension. This decomposition is an analogue of well-known constructions for divisors. Examples illustrate how Zariski decompositions of cycle classes reflect the geometry of the underlying space. We also analyze the birational behavior of Zariski decompositions, leading to a Fujita approximation-type result for curve classes.

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arXiv2016

Kernels of numerical pushforwards

Let $π: X \to Y$ be a morphism of projective varieties and consider the pushforward map $π_*: N_k(X) \to N_k(Y)$ of numerical cycle classes. We show that when the Chow groups of points of the fibers are as simple as they can be, then the kernel of $π_*$ is spanned by k-cycles contracted by $π$.

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arXiv2016

Positive cones of dual cycle classes

We study generalizations for higher codimension cycles of several well-known definitions of the nef cone of divisors on a projective variety. These generalizations fix some of the pathologies exhibited by the classical nef cone of higher codimension classes. As an application, we recover the expected properties of the pseudoeffective cones $\overline{Eff}_{k}(X)$ for all k.

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arXiv2016

Morphisms and faces of pseudo-effective cones

Let $π: X \to Y$ be a morphism of projective varieties and suppose that $α$ is a pseudo-effective numerical cycle class satisfying $π_*α= 0$. A conjecture of Debarre, Jiang, and Voisin predicts that $α$ is a limit of classes of effective cycles contracted by $π$. We establish new cases of the conjecture for higher codimension cycles. In particular we prove a strong version when $X$ is a fourfold and $π$ has relative dimension one.

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arXiv2015

Volume and Hilbert function of R-divisors

If D is a big divisor and E is an effective divisor, then vol(D-E)<=vol(D)<=vol(D+E). We discuss when each inequality is an equality. Surprisingly, the answer is that the asymptotic equality vol(D-E)=vol(D) is equivalent to the equality of the Hilbert functions of D-E and D. The analogous result holds when vol(D)=vol(D+E).

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arXiv2012

Asymptotic Schur decomposition of Veronese syzygy functors

The syzygies of the d-th Veronese embedding of $\mathbb P(V)$ are functors of the complex vector space V. From a certain perspective, we show that as d grows, their Schur functor decomposition is very rich whenever they are not zero. This is deduced from an asymptotic study of related plethysms. We also obtain other results related to a question of Ein and Lazarsfeld.

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arXiv2012

The cones of effective cycles on projective bundles over curves

Generalizing work done by Miyaoka and others in the case of divisors and of curves, we compute the cones of effective cycles of arbitrary dimension on a projective bundle over a complex projective curve in terms of the numerical data in an associated Harder-Narasimhan filtration. An application to cycles on projective bundles over a smooth complex projective base of arbitrary dimension is also given.

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arXiv2011

Local volumes on normal algebraic varieties

In this paper we study a notion of volume for Cartier divisors on arbitrary blow-ups of normal complex algebraic varieties of dimension greater than one, with a distinguished point. We apply this to study a volume for normal isolated singularities, generalizing work on surfaces done by Wahl. We also compare this volume of isolated singularities to a different generalization due to Boucksom, de Fernex and Favre.

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