Pietro De Poi
In this article we study congruences of lines in $\mathbb{P}^n$, and in particular of order one. After giving general results, we obtain a complete classification in the case of $\mathbb{P}^4$ in which the fundamental surface $F$ is in fact a variety-i.e. it is integral-and the congruence is the irreducible set of the trisecant lines of $F$.
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A congruence is a surface in the Grassmannian ${\rm Gr}(2, 4)$. In this paper, we consider the normalization of congruence of bitangents to a hypersurface in $\mathbb P^3$. We call it the Fano congruence of bitangents. We give a criterion for smoothness of the Fano congruence of bitangents and describe explicitly their degenerations in a general Lefschetz pencil in the space of hypersurfaces in $\mathbb P^3$.
The existence is proved of two new families of locally Cohen-Macaulay sextic threefolds in $\mathbb{P}^5$, which are not quadratically normal. These threefolds arise naturally in the realm of first order congruences of lines as focal loci and in the study of the completely exceptional Monge-Ampère equations. One of these families comes from a smooth congruence of multidegree $(1,3,3)$ which is a smooth Fano fourfold of index two and genus 9.
In this article we obtain a complete description of the congruences of lines in $\p^4$ of order one provided that the fundamental surface $F$ is non-reduced (and possibly reducible) at one of its generic points, and their classification under the hypothesis that $(F)_{\red}$ is smooth.
Strong congruence spaces and dimension in $\mathbb{F}_1$-geometry
本文提出强同余空间作为幺半群概形的维数概念，并证明代数闭基上环面幺半群概形的强同余空间与对应复环面簇维数相同。
A congruence is a surface in the Grassmannian $\mathrm{Gr}(1,\mathbb{P}^3)$ of lines in projective $3$-space. To a space curve $C$, we associate the Chow hypersurface in $\mathrm{Gr}(1,\mathbb{P}^3)$ consisting of all lines which intersect $C$. We compute the singular locus of this hypersurface, which contains the congruence of all secants to $C$. A surface $S$ in $\mathbb{P}^3$ defines the Hurwitz hypersurface in $\mathrm{Gr}(1,\mathbb{P}^3)$ of all lines which are tangent to $S$. We show that its singular locus has two components for general enough $S$: the congruence of bitangents and the c…
The works of Hassett and Kuznetsov identify countably many divisors $C_d$ in the open subset of $\mathbb{P}^{55}=\mathbb{P}(H^0(\mathcal{O}_{\mathbb{P}^5}(3)))$ parametrizing all cubic 4-folds and conjecture that the cubics corresponding to these divisors are precisely the rational ones. Rationality has been known classically for the first family $C_{14}$. We use congruences of 5-secant conics to prove rationality for the first three of the families $C_d$, corresponding to $d=14, 26, 38$ in Hassett's notation.
In this article we obtain the classification of the congruences of lines with one-dimensional focal locus. It turns out that one can restrict to study the case of $\mathbb{P}^3$.
In this note we present a notion of fundamental scheme for Cohen- Macaulay, order 1, irreducible congruences of lines. We show that such a congruence is formed by the k-secant lines to its fundamental scheme for a number k that we call the secant index of the congruence. If the fundamental scheme X is a smooth connected variety in PN, then k = (N-1)/(c-1) (where c is the codimension of X) and X comes equipped with a special tangency divisor cut out by a virtual hypersurface of degree k-2 (to be precise, linearly equivalent to a section by an hypersurface of degree (k-2) without being cut by on…
This note is an appendix to 'Measures of irrationality for hypersurfaces of large degree' by L. Ein, R. Lazarsfeld and B. Ullery. We prove an existence result for families of curves having low gonality, and lying on fundamental loci of first order congruences of lines in $\mathbb{P}^{n+1}$. As an application, we follow the ideas of the main paper, and we present a slight refinement of a theorem included in it. In particular, we show that given a very general hypersurface $X\subset \mathbb{P}^{n+1}$ of degree $d \geq 3n-2 \geq 7$, and a dominant rational map $f \colon X \dashrightarrow \mathbb{…
We study congruences of lines $X_ω$ defined by a sufficiently general choice of an alternating 3-form $ω$ in $n+1$ dimensions, as Fano manifolds of index $3$ and dimension $n-1$. These congruences include the $\mathrm{G}_2$-variety for $n=6$ and the variety of reductions of projected $\mathbb{P}^2 \times \mathbb{P}^2$ for $n=7$. We compute the degree of $X_ω$ as the $n$-th Fine number and study the Hilbert scheme of these congruences proving that the choice of $ω$ bijectively corresponds to $X_ω$ except when $n=5$. The fundamental locus of the congruence is also studied together with its singu…
We study a certain family of determinantal quintic hypersurfaces in $\mathbb{P}^{4}$ whose singularities are similar to the well-studied Barth-Nieto quintic. Smooth Calabi-Yau threefolds with Hodge numbers $(h^{1,1},h^{2,1})=(52,2)$ are obtained by taking crepant resolutions of the singularities. It turns out that these smooth Calabi-Yau threefolds are in a two dimensional mirror family to the complete intersection Calabi-Yau threefolds in $\mathbb{P}^{4}\times\mathbb{P}^{4}$ which have appeared in our previous study of Reye congruences in dimension three. We compactify the two dimensional fam…
We present a new framework for multi-view geometry in computer vision. A camera is a mapping between $\mathbb{P}^3$ and a line congruence. This model, which ignores image planes and measurements, is a natural abstraction of traditional pinhole cameras. It includes two-slit cameras, pushbroom cameras, catadioptric cameras, and many more. We study the concurrent lines variety, which consists of $n$-tuples of lines in $\mathbb{P}^3$ that intersect at a point. Combining its equations with those of various congruences, we derive constraints for corresponding images in multiple views. We also study …
In this article we classify all the smooth threefolds of P^5 with an apparent quadruple point provided that the family of its 4-secant lines is an irreducible (first order) congruence. This is sufficient to conclude the classification of all the smooth codimension two varieties of P^n with one apparent (n-1)-point and with irreducible family of (n-1)-secant lines.
Algebra and Geometry of Irreducible toric vector bundles of rank $n$ on $\mathbb{P}^n$
本文构造了 $\mathbb{P}^n$ 上秩 $n$ 不可约环面向量丛射影化的 Cox 环表示，并据此证明其总满足 Fujita 自由性与丰富性猜想。
We study general properties of the dessins d'enfants associated with the Hecke congruence subgroups $Γ_0(N)$ of the modular group $\mathrm{PSL}_2(\mathbb{R})$. The definition of the $Γ_0(N)$ as the stabilisers of couples of projective lattices in a two-dimensional vector space gives an interpretation of the quotient set $Γ_0(N)\backslash\mathrm{PSL}_2(\mathbb{R})$ as the projective lattices $N$-hyperdistant from a reference one, and hence as the projective line over the ring $\mathbb{Z}/N\mathbb{Z}$. The natural action of $\mathrm{PSL}_2(\mathbb{R})$ on the lattices defines a dessin d'enfant s…
Hecke algebras for the 1st congruence subgroup and bundles on ${\mathbb P}^1$ I: the case of finite field
本文证明有限域上射影直线带两端平凡化的 $G$-丛有限支撑函数空间同构于第一同余子群 Hecke 代数的正则双模，并刻画其稳定中心与尖点表示提升特征。
Let $ω=(-1+\sqrt{-3})/2$. For any lattice $P\subseteq \mathbb{Z}^n$, $\mathcal{P}=P+ωP$ is a subgroup of $\mathcal{O}_K^n$, where $\mathcal{O}_K=\mathbb{Z}[ω]\subseteq \mathbb{C}$. As $\mathbb{C}$ is naturally isomorphic to $\mathbb{R}^2$, $\mathcal{P}$ can be regarded as a lattice in $\mathbb{R}^{2n}$. Let $P$ be a multiplicative lattice (principal lattice or congruence lattice) introduced by Rosenbloom and Tsfasman. We concatenate a family of special codes with $t_{\mathfrak{P}}^\ell\cdot(P+ωP)$, where $t_{\mathfrak{P}}$ is the generator of a prime ideal $\mathfrak{P}$ of $\mathcal{O}_K$. Ap…
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