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Vincenzo Di Gennaro

Publications and source records attributed to Vincenzo Di Gennaro.

At least 19 recordsLinked to original sources
arXiv2026

On the genus of a curve in a projective $3$-fold

Let $X\subset \mathbb P^r$ be a projective factorial variety of dimension $3$, degree $n$, with at worst isolated singularities. Assume that the Picard group of $X$ is generated by the hyperplane section class. Let $C\subset X$ be a projective subscheme of dimension $1$, degree $d\gg n$, and arithmetic genus $p_a(C)$. Improving a recent result by Liu, we exhibit a Castelnuovo's bound for $p_a(C)$. In the case $X$ is Calabi-Yau, our bound gives a step forward for a certain conjecture concerning the vanishing of Gopakumar-Vafa invariants of $X$.

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arXiv2024

On the genus of projective curves not contained in hypersurfaces of given degree, II

Fix integers $r\geq 4$ and $i\geq 2$. Let $C$ be a non-degenerate, reduced and irreducible complex projective curve in $\mathbb P^r$, of degree $d$, not contained in a hypersurface of degree $\leq i$. Let $p_a(C)$ be the arithmetic genus of $C$. Continuing previous research, under the assumption $d\gg \max\{r,i\}$, in the present paper we exhibit a Castelnuovo bound $G_0(r;d,i)$ for $p_a(C)$. In general, we do not know whether this bound is sharp. However, we are able to prove it is sharp when $i=2$, $r=6$ and $d\equiv 0,3,6$ (mod $9$). Moreover, when $i=2$, $r\geq 9$, $r$ is divisible by $3$, and $d\equiv 0$ (mod $r(r+3)/6$), we prove that if $G_0(r;d,i)$ is not sharp, then for the maximal value of $p_a(C)$ there are only three possibilities. The case in which $i=2$ and $r$ is not divisible by $3$ has already been examined in the literature. We give some information on the extremal curves.

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arXiv2022

Nilpotent cone and bivariant theory

We exhibit a new proof, relying on bivariant theory, that the nilpotent cone is rationally smooth. Our approach enables us to prove a slightly more general statement.

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arXiv2022

On the genus of projective curves not contained in hypersurfaces of given degree

Fix integers $r\geq 4$ and $i\geq 2$ (for $r=4$ assume $i\geq 3$). Assuming that the rational number $s$ defined by the equation $\binom{i+1}{2}s+(i+1)=\binom{r+i}{i}$ is an integer, we prove an upper bound for the genus of a reduced and irreducible complex projective curve in $\mathbb P^r$, of degree $d\gg s$, not contained in hypersurfaces of degree $\leq i$. It turns out that this bound coincides with the Castelnuovo's bound for a curve of degree $d$ in $\mathbb P^{s+1}$. We prove that the bound is sharp if and only if there exists an integral surface $S\subset \mathbb P^r$ of degree $s$, not contained in hypersurfaces of degree $\leq i$. Such a surface, if existing, is necessarily the isomorphic projection of a rational normal scroll surface of degree $s$ in $\mathbb P^{s+1}$. The existence of such a surface $S$ is known for $i=2$ and $i=3$. It follows that, when $i=2$ or $i=3$, the bound is sharp, and the extremal curves are isomorphic projection in $\mathbb P^r$ of Castelnuovo's curves of degree $d$ in $\mathbb P^{s+1}$.

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arXiv2022

The genus of curves in $\mathbb P^4$ and $\mathbb P^5$ not contained in quadrics

A classical problem in the theory of projective curves is the classification of all their possible genera in terms of the degree and the dimension of the space where they are embedded. Fixed integers $r,d,s$, Castelnuovo-Halphen's theory states a sharp upper bound for the genus of a non-degenerate, reduced and irreducible curve of degree $d$ in $\mathbb P^r$, under the condition of being not contained in a surface of degree $<s$. This theory can be generalized in several ways. For instance, fixed integers $r,d,k$, one may ask for the maximal genus of a curve of degree $d$ in $\mathbb P^r$, not contained in a hypersurface of a degree $<k$. In the present paper we examine the genus of curves $C$ of degree $d$ in $\mathbb P^r$ not contained in quadrics (i.e. $h^0(\mathbb P^r, \mathcal I_C(2))=0$). When $r=4$ and $r=5$, and $d\gg0$, we exhibit a sharp upper bound for the genus. For certain values of $r\geq 7$, we are able to determine a sharp bound except for a constant term, and the argument applies also to curves not contained in cubics.

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arXiv2021

Bivariant class of degree one

Let $f:X\to Y$ be a projective birational morphism, between complex quasi-projective varieties. Fix a bivariant class $θ\in H^0(X\stackrel{f}\to Y)\cong Hom_{D^{b}_{c}(Y)}(Rf_*\mathbb A_X, \mathbb A_Y)$ (here $\mathbb A$ is a Noetherian commutative ring with identity, and $\mathbb A_X$ and $\mathbb A_Y$ denote the constant sheaves). Let $θ_0:H^0(X)\to H^0(Y)$ be the induced Gysin morphism. We say that {\it $θ$ has degree one} if $θ_0(1_X)= 1_Y\in H^0(Y)$. This is equivalent to say that $θ$ is a section of the pull-back $f^*: \mathbb A_Y\to Rf_*\mathbb A_X$, i.e. $θ\circ f^*={\text{id}}_{\mathbb A_Y}$, and it is also equivalent to say that $\mathbb A_Y$ is a direct summand of $Rf_*\mathbb A_X$. We investigate the consequences of the existence of a bivariant class of degree one. We prove explicit formulas relating the (co)homology of $X$ and $Y$, which extend the classic formulas of the blowing-up. These formulas are compatible with the duality morphism. Using which, we prove that the existence of a bivariant class $θ$ of degree one for a resolution of singularities, is equivalent to require that $Y$ is an $\mathbb A$-homology manifold. In this case $θ$ is unique, and the Betti numbers of the singular locus ${\text{Sing}}(Y)$ of $Y$ are related with the ones of $f^{-1}({\text{Sing}}(Y))$.

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arXiv2021

A lower bound for $χ(\mathcal O_S)$

Let $(S,\mathcal L)$ be a smooth, irreducible, projective, complex surface, polarized by a very ample line bundle $\mathcal L$ of degree $d > 25$. In this paper we prove that $χ(\mathcal O_S)\geq -\frac{1}{8}d(d-6)$. The bound is sharp, and $χ(\mathcal O_S)=-\frac{1}{8}d(d-6)$ 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 $T$. Moreover, this is equivalent to the fact that the general hyperplane section $H\in |H^0(S,\mathcal L)|$ of $S$ is the projection of a curve $C$ contained in the Veronese surface $V\subseteq \mathbb P^5$, from a point $x\in V\backslash C$.

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arXiv2020

Intersection cohomology and Severi's varieties

Let $X^{2n}\subseteq \mathbb{P} ^N$ be a smooth projective variety. Consider the intersection cohomology complex of the local system $R^{2n-1}π{_*}\mathbb{Q}$, where $π$ denotes the projection from the universal hyperplane family of $X^{2n}$ to ${(\mathbb{P} ^N)}^{\vee}$. We investigate the cohomology of the intersection cohomology complex $IC(R^{2n-1}π{_*}\mathbb{Q})$ over the points of a Severi's variety, parametrizing nodal hypersurfaces, whose nodes impose independent conditions on the very ample linear system giving the embedding in $\mathbb{P} ^N$.

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arXiv2020

On the topology of a resolution of isolated singularities, II

Let $Y$ be a complex projective variety of dimension $n$ with isolated singularities, $π:X\to Y$ a resolution of singularities, $G:=π^{-1}\left(\rm{Sing}(Y)\right)$ the exceptional locus. From the Decomposition Theorem one knows that the map $H^{k-1}(G)\to H^k(Y,Y\backslash {\rm{Sing}}(Y))$ vanishes for $k>n$. It is also known that, conversely, assuming this vanishing one can prove the Decomposition Theorem for $π$ in few pages. The purpose of the present paper is to exhibit a direct proof of the vanishing. As a consequence, it follows a complete and short proof of the Decomposition Theorem for $π$, involving only ordinary cohomology.

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arXiv2019

A remark on the genus of curves in $\mathbf P^4$

Let $C$ be an irreducible, reduced, non-degenerate curve, of arithmetic genus $g$ and degree $d$, in the projective space $\mathbf P^4$ over the complex field. Assume that $C$ satisfies the following {\it flag condition of type $(s,t)$}: {$C$ does not lie on any surface of degree $<s$, and on any hypersurface of degree $<t$}. Improving previous results, in the present paper we exhibit a Castelnuovo-Halphen type bound for $g$, under the assumption $s\leq t^2-t$ and $d\gg t$. In the range $t^2-2t+3\leq s\leq t^2-t$, $d\gg t$, we are able to give some information on the extremal curves. They are arithmetically Cohen-Macaulay curves, and lie on a flag like $S\subset F$, where $S$ is a surface of degree $s$, $F$ a hypersurface of degree $t$, $S$ is unique, and its general hyperplane section is a space extremal curve, not contained in any surface of degree $<t$. In the case $d\equiv 0$ (modulo $s$), they are exactly the complete intersections of a surface $S$ as above, with a hypersurface. As a consequence of previous results, we get a bound for the speciality index of a curve satisfying a flag condition.

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arXiv2018

On a resolution of singularities with two strata

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.

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arXiv2017

On the topology of a resolution of isolated singularities

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.

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arXiv2016

A speciality Theorem for curves in $\mathbb{P}^5$ contained in Noether-Lefschetz general fourfolds

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 ${\frac{d}{snk}}$.

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arXiv2016

A lower bound for $K^2_S$

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 $T$.

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arXiv2016

Néron-Severi group of a general hypersurface

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.

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arXiv2014

Algebraic versus homological equivalence for singular varieties

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.

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arXiv2012

Noether-Lefschetz Theory with base locus

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 $Z$.

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