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Daniel Naie

Publications and source records attributed to Daniel Naie.

7 recordsLinked to original sources
arXiv2016

Surfaces in $\mathbb{P}^4$ lying on small degree hypersurfaces

Since the work of Ellingsrud and Peskine at the end of 1980s, it has been known that, with the exception of a finite number of families, smooth compact complex surfaces in $\mathbb{P}^4$ with prescribed Chern classes must lie on hypersurfaces of degree $m\leq 5$. The study of surfaces lying on a small degree hypersurface in $\mathbb{P}^4$---small meaning $\leq5$---seems to be a way of obtaining empirical data leading to a better conceptual understanding of surfaces in $\mathbb{P}^4$. From this perspective, two main issues are considered in the paper: - an analogue of the Hartshorne-Lichtenbaum finiteness results for smooth surfaces of general type contained in a small degree hypersurface in $\mathbb{P}^4$, - a study of the irregularity of smooth surfaces contained in a small degree hypersurface in $\mathbb{P}^4$.

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arXiv2011

Twisted Kodaira-Spencer classes and the geometry of surfaces of general type

We study the cohomology groups $H^1(X,Θ_X(-mK_X))$, for $m\geq1$, where $X$ is a smooth minimal complex surface of general type, $Θ_X$ its holomorphic tangent bundle, and $K_X$ its canonical divisor. One of the main results is a precise vanishing criterion for $H^1(X,Θ_X (-K_X))$. The proof is based on the geometric interpretation of non-zero cohomology classes of $H^1(X,Θ_X (-K_X))$. This interpretation in turn uses higher rank vector bundles on $X$. We apply our methods to the long standing conjecture saying that the irregularity of surfaces in $\PP^4$ is at most 2. We show that if $X$ has prescribed Chern numbers, no irrational pencil, and is embedded in $\PP^4$ with a sufficiently large degree, then the irregularity of $X$ is at most 3.

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arXiv2008

Jumping numbers of a unibranch curve on a smooth surface

A formula for the jumping numbers of a curve unibranch at a singular point is established. The jumping numbers are expressed in terms of the Enriques diagram of the log resolution of the singularity, or equivalently in terms of the canonical set of generators of the semigroup of the curve at the singular point.

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arXiv2007

Log-canonical threshold for curves on a smooth surface

It is shown that the log-canonical threshold of a curve with an isolated singularity is computed by the term ideal of the curve in a suitable system of local parameters at the singularity. The proof uses the Enriques diagram of the singularity and shows that the log-canonical threshold depends only on a non-degenerate path of that diagram.

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arXiv2006

The irregularity of cyclic multiple planes after Zariski

A formula for the irregularity of a cyclic multiple plane associated to a branch curve that has arbitrary singularities and is transverse to the line at infinity is established. The irregularity is expressed as a sum of superabundances of linear systems associated to some multiplier ideals of the branch curve and the proof rests on the theory of standard cyclic coverings. Explicit computations of multiplier ideals are performed and some applications are presented.

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arXiv2003

Rationality properties of manifolds containing quasi-lines

Let X be a complex, rationally connected, projective manifold. We show that X admits a modification X' that contains a quasi-line, ie a smooth rational curve whose normal bundle is a direct sum of copies of O_{P^1}(1). For manifolds containing quasi-lines, a sufficient condition of rationality is exploited: There is a unique quasi-line from a given family passing through two general points. We define a numerical birational invariant, e(X), and prove that X is rational if and only if e(X)=1. If X is rational, there is a modification X' which is strongly-rational We prove that strongly-rational varieties are stable under smooth, small deformations. Finally, we relate the previous results and formal geometry. This relies on \tilde{e}(X,Y), a numerical invariant of a given quasi-line Y that depends only on the formal completion of X along Y. As applications we show various instances in which X is determined by this formal completion. We also formulate a basic question about the birational invariance of \tilde{e}(X,Y).

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