arXiv subjects
Publications and source records attributed to Davide Franco.
We give a short and self-contained proof of the Decomposition Theorem for the non-small resolution of a Special Schubert variety. We also provide an explicit description of the perverse cohomology sheaves. As a by-product of our approach, we obtain a simple proof of the Relative Hard Lefschetz Theorem.
A classical approach to investigate a closed projective scheme $W$ consists of considering a general hyperplane section of $W$, which inherits many properties of $W$. The inverse problem that consists in finding a scheme $W$ starting from a possible hyperplane section $Y$ is called a {\em lifting problem}, and every such scheme $W$ is called a {\em lifting} of $Y$. Investigations in this topic can produce methods to obtain schemes with specific properties. For example, any smooth point for $Y$ is smooth also for $W$. We characterize all the liftings of $Y$ with a given Hilbert polynomial by a parameter scheme that is obtained by gluing suitable affine open subschemes in a Hilbert scheme and is described through the functor it represents. We use constructive methods from Gröbner and marked bases theories. Furthermore, by classical tools we obtain an analogous result for equidimensional liftings. Examples of explicit computations are provided.
Let $X$ be a complex, irreducible, quasi-projective variety, and $π:\widetilde X\to X$ a resolution of singularities of $X$. Assume that the singular locus ${\text{Sing}}(X)$ of $X$ is smooth, that the induced map $π^{-1}({\text{Sing}}(X))\to {\text{Sing}}(X)$ is a smooth fibration admitting a cohomology extension of the fiber, and that $π^{-1}({\text{Sing}}(X))$ has a negative normal bundle in $\widetilde X$. We present a very short and explicit proof of the Decomposition Theorem for $π$, providing a way to compute the intersection cohomology of $X$ by means of the cohomology of $\widetilde X$ and of $π^{-1}({\text{Sing}}(X))$. Our result applies to special Schubert varieties with two strata, even if $π$ is non-small. And to certain hypersurfaces of $\mathbb P^5$ with one-dimensional singular locus.
Let $Y$ be a complex projective variety of dimension $n$ with isolated singularities, $π:X\to Y$ a resolution of singularities, $G:=π^{-1}{\rm{Sing}}(Y)$ the exceptional locus. From Decomposition Theorem one knows that the map $H^{k-1}(G)\to H^k(Y,Y\backslash {\rm{Sing}}(Y))$ vanishes for $k>n$. Assuming this vanishing, we give a short proof of Decomposition Theorem for $π$. A consequence is a short proof of the Decomposition Theorem for $π$ in all cases where one can prove the vanishing directly. This happens when either $Y$ is a normal surface, or when $π$ is the blowing-up of $Y$ along ${\rm{Sing}}(Y)$ with smooth and connected fibres, or when $π$ admits a natural Gysin morphism. We prove that this last condition is equivalent to say that the map $H^{k-1}(G)\to H^k(Y,Y\backslash {\rm{Sing}}(Y))$ vanishes for any $k$, and that the pull-back $π^*_k:H^k(Y)\to H^k(X)$ is injective. This provides a relationship between Decomposition Theorem and Bivariant Theory.
Let $C\subset \mathbb P^r$ be an integral projective curve. We define the speciality index $e(C)$ of $C$ as the maximal integer $t$ such that $h^0(C,ω_C(-t))>0$, where $ω_C$ denotes the dualizing sheaf of $C$. In the present paper we consider $C\subset \mathbb P^5$ an integral degree $d$ curve and we denote by $s$ the minimal degree for which there exists a hypersurface of degree $s$ containing $C$. We assume that $C$ is contained in two smooth hypersurfaces $F$ and $G$, with $deg(F)=n>k=deg (G)$. We assume additionally that $F$ is Noether-Lefschetz general, i.e. that the $2$-th Néron-Severi group of $F$ is generated by the linear section class. Our main result is that in this case the speciality index is bounded as $e(C)\leq {\frac{d}{snk}}+s+n+k-6.$ Moreover equality holds if and only if $C$ is a complete intersection of $T:=F\cap G$ with hypersurfaces of degrees $s$ and ${\frac{d}{snk}}$.
In this paper we characterize the Blowing-up maps of ordinary singularities for which there exists a natural Gysin morphism, i.e. a bivariant class $θ\in Hom_{D(Y)}(Rπ_*\mathbb Q_X, \mathbb Q_Y)$, compatible with pullback and with restriction to the complement of the singularity.
We define an adelic version of a CM elliptic curve $E$ which is equipped with an action of the profinite completion of the endomorphism ring of $E$. The adelic elliptic curve so obtained is provided with a natural embedding into the adelic Heisenberg group. We embed into the adelic Heisenberg group the set of commensurability classes of arithmetic $1$-dimensional $\mathbb{K}$-lattices (here and subsequently, $\mathbb{K}$ denotes a quadratic imaginary number field) and define theta functions on it. We also embed the groupoid of commensurability modulo dilations into the union of adelic Heisenberg groups relative to a set of representatives of elliptic curves with $R$-multiplication ($R$ is the ring of algebraic integers of $\mathbb{K}$). We thus get adelic theta functions on the set of $1$-dimensional $\mathbb{K}$-lattices and on the groupoid of commensurability modulo dilations. Adelic theta functions turn out to be acted by the adelic Heisenberg group and behave nicely under complex automorphisms (Theorems 6.12 and 6.14).
We describe how to obtain the imprimitivity bimodules of the noncommutative torus from a "principal bundle" construction, where the total space is a quasi-associative deformation of a 3-dimensional Heisenberg manifold.
Let $(S,\mathcal L)$ be a smooth, irreducible, projective, complex surface, polarized by a very ample line bundle $\mathcal L$ of degree $d > 35$. In this paper we prove that $K^2_S\geq -d(d-6)$. The bound is sharp, and $K^2_S=-d(d-6)$ if and only if $d$ is even, the linear system $|H^0(S,\mathcal L)|$ embeds $S$ in a smooth rational normal scroll $T\subset \mathbb P^5$ of dimension $3$, and here, as a divisor, $S$ is linearly equivalent to $\frac{d}{2}Q$, where $Q$ is a quadric on $T$.
In this paper we extend the well known theorem of Angelo Lopez concerning the Picard group of the general space projective surface containing a given smooth projective curve, to the intermediate Néron-Severi group of a general hypersurface in any smooth projective variety.
Using the Weil-Brezin-Zak transform of solid state physics, we describe line bundles over elliptic curves in terms of Weyl operators. We then discuss the connection with finitely-generated projective modules over the algebra $A_θ$ of the noncommutative torus. We show that such $A_θ$-modules have a natural interpretation as Moyal deformations of vector bundles over an elliptic curve $E_τ$, under the condition that the deformation parameter $θ$ and the modular parameter $τ$ satisfy a non-trivial relation.
Let $ Y \subseteq \Bbb P^N $ be a possibly singular projective variety, defined over the field of complex numbers. Let $X$ be the intersection of $Y$ with $h$ general hypersurfaces of sufficiently large degrees. Let $d>0$ be an integer, and assume that $\dim Y=n+h$ and $ \dim Y_{sing} \le \min\{ d+h-1 , n-1 \} $. Let $Z$ be an algebraic cycle on $Y$ of dimension $d+h$, whose homology class in $H_{2(d+h)}(Y; \Bbb Q)$ is non-zero. In the present paper we prove that the restriction of $Z$ to $X$ is not algebraically equivalent to zero. This is a generalization to the singular case of a result due to Nori in the case $Y$ is smooth. As an application we provide explicit examples of singular varieties for which homological equivalence is different from the algebraic one.
Let $Z$ be a closed subscheme of a smooth complex projective variety $Y\subseteq \Ps^N$, with $\dim\,Y=2r+1\geq 3$. We describe the intermediate Néron-Severi group (i.e. the image of the cycle map $A_r(X)\to H_{2r}(X;\mathbb{Z})$) of a general smooth hypersurface $X\subset Y$ of sufficiently large degree containing $Z$.
Several observed anomalies in neutrino oscillation data can be explained by a hypothetical fourth neutrino separated from the three standard neutrinos by a squared mass difference of a few eV^2. We show that this hypothesis can be tested with a PBq (ten kilocurie scale) 144Ce or 106Ru antineutrino beta-source deployed at the center of a large low background liquid scintillator detector. In particular, the compact size of such a source could yield an energy-dependent oscillating pattern in event spatial distribution that would unabiguously determine neutrino mass differences and mixing angles.
Fix integers $r,d,s,π$ with $r\geq 4$, $d\gg s$, $r-1\leq s \leq 2r-4$, and $π\geq 0$. Refining classical results for the genus of a projective curve, we exhibit a sharp upper bound for the arithmetic genus $p_a(C)$ of an integral projective curve $C\subset {\mathbb{P}^r}$ of degree $d$, assuming that $C$ is not contained in any surface of degree $ π$. Next we discuss other types of bound for $p_a(C)$, involving conditions on the entire Hilbert polynomial of the integral surfaces on which $C$ may lie.
By the fundamental work of Griffiths one knows that, under suitable assumption, homological and algebraic equivalence do not coincide for a general hypersurface section of a smooth projective variety $Y$. In the present paper we prove the same result in case $Y$ has isolated singularities.
Let $Z$ be a closed subscheme of a smooth complex projective complete intersection variety $Y\subseteq \Ps^N$, with $dim Y=2r+1\geq 3$. We describe the Néron-Severi group $NS_r(X)$ of a general smooth hypersurface $X\subset Y$ of sufficiently large degree containing $Z$.
Let $Y$ be an $(m+1)$-dimensional irreducible smooth complex projective variety embedded in a projective space. Let $Z$ be a closed subscheme of $Y$, and $δ$ be a positive integer such that $\mathcal I_{Z,Y}(δ)$ is generated by global sections. Fix an integer $d\geq δ+1$, and assume the general divisor $X \in |H^0(Y,\ic_{Z,Y}(d))|$ is smooth. Denote by $H^m(X;\mathbb Q)_{\perp Z}^{\text{van}}$ the quotient of $H^m(X;\mathbb Q)$ by the cohomology of $Y$ and also by the cycle classes of the irreducible components of dimension $m$ of $Z$. In the present paper we prove that the monodromy representation on $H^m(X;\mathbb Q)_{\perp Z}^{\text{van}}$ for the family of smooth divisors $X \in |H^0(Y,\ic_{Z,Y}(d))|$ is irreducible.