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Maria Chiara Brambilla

Publications and source records attributed to Maria Chiara Brambilla.

At least 19 recordsLinked to original sources
arXiv2026

On the strong base locus of a projective variety

We introduce and study the base locus and the strong base locus of a projective variety X. The base locus of X parametrizes configurations of smooth points of X where the span of the tangent spaces of X at these points intersects X at some additional smooth point. The strong base locus parametrizes configurations of smooth points of X for which the span of the tangent spaces of X at the given configuration contains the entire tangent space at an additional point. These notions originate from the study of base loci of tangential projections, are strictly related to interpolation problems with double points in special position, and provide a natural framework to study tangential contact for nongeneral points. We give first properties and explore connections with Terracini loci and with the concept of identifiability. We focus on tensor-related varieties and characterize the nonemptiness of base loci and strong base loci for Veronese and Segre-Veronese varieties.

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arXiv2025

Terracini loci and a codimension one Alexander-Hirschowitz theorem

The Terracini locus $\mathbb{T}(n, d; x)$ is the locus of all finite subsets $S$ of $ \mathbb{P}^n$ of cardinality $x$ such that $\langle S \rangle = \mathbb{P}^n$, $h^0(\mathcal{I}_{2S}(d)) > 0$, and $h^1(\mathcal{I}_{2S}(d)) > 0$. The celebrated Alexander-Hirschowitz Theorem classifies the triples $(n,d,x)$ for which $\dim\mathbb{T}(n, d; x)=xn$. Here we fully characterize the next step in the case $n=2$, namely, we prove that $\mathbb{T}(2,d;x)$ has at least one irreducible component of dimension $2x-1$ if and only if either $(d,x)\in\{(4,4),(4,6),$ $(5,6),(5,7),$ $(6,9),(6,10)\}$, or $d\ge 7$, $d\equiv 1,2 \pmod{3}$ and $x=(d+2)(d+1)/6$.

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arXiv2025

Birational geometry of blowups via Weyl chamber decompositions and actions on curves

We study the birational geometry of $X^n_s$, the blow-up of $\mathbb{P}^n_\mathbb{C}$ at $s$ points in general position. We identify a set of subvarieties, which we call Weyl $r$-planes, that belong to an orbit for the action of the Weyl group on $r$-cycles. They satisfy the following properties: they appear as stable base locus of divisors; each Weyl $r$-plane is swept out by an $(n-r)$-moving curve class; moreover, if $s\ge n+3$, for any fixed $r$ all these curve classes belong to the same orbit for the Weyl action. For Mori dream spaces of type $X^n_s$, all such orbits are finite and they allow to reinterpret Mukai's description of the Mori chamber decomposition of the effective cone in terms of $(n-r)$-moving curve classes, unifying previous different approaches. If $X^n_s$ is not a Mori dream space, there are infinitely many Weyl $r$-planes. These yields the definition of the Weyl chamber decomposition of the pseudoeffective cone of divisors. We pose the question as to whether the nef chamber decomposition can be defined (in the negative part of $\overline{\mathrm{Eff}}(X^n_s)$) and, if this is the case, whether it coincides with the Weyl chamber decomposition. We conjecture that the answer is affirmative for $X^3_8$ and $X^5_9$.

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arXiv2025

Duality and polyhedrality of cones for Mori dream spaces

Our goal is twofold. On one hand we show that the cones of divisors ample in codimension $k$ on a Mori dream space are rational polyhedral. On the other hand we study the duality between such cones and the cones of $k$-moving curves by means of the Mori chamber decomposition of the former. We give a new proof of the weak duality property (already proved by Payne and Choi) and we exhibit an interesting family of examples for which strong duality holds.

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arXiv2024

Non-defectivity of Segre-Veronese varieties

We prove that Segre-Veronese varieties are never secant defective if each degree is at least three. The proof is by induction on the number of factors, degree and dimension. As a corollary, we give an almost optimal non-defectivity result for Segre-Veronese varieties with one degree equal to one and all the others at least three.

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arXiv2024

Nonvanishing and Abundance for cones of movable divisors

Let $\overline{\mathrm{Mov}}^k(X)$ be the closure of the cone $\mathrm{Mov}^k(X)$ generated by classes of effective divisors on a projective variety $X$ with stable base locus of codimension at least $k+1$. We propose a generalized version of the Log Nonvanishing Conjecture and of the Log Abundance Conjecture for a klt pair $(X,Δ)$, that is: if $K_X+Δ\in \overline{\mathrm{Mov}}^{k}(X)$, then $K_X+Δ\in \mathrm{Mov}^{k}(X)$. Moreover, we prove that if the Log Minimal Model Program, the Log Nonvanishing, and the Log Abundance hold, then so does our conjecture.

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arXiv2024

Minimal Terracini loci in projective spaces

We characterize the number of points for which there exist non-empty Terracini sets of points in $\mathbb{P}^n$. Then we study minimally Terracini finite sets of points in $\mathbb{P}^n$ and we obtain a complete description in the case of $\mathbb{P}^3$, when the number of points is less than twice the degree of the linear system.

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arXiv2023

Twistor fibers in hypersurfaces of the flag threefold

We study surfaces of bidegree (1,d) contained in the flag threefold in relation to the twistor projection. In particular, we focus on the number and the arrangement of twistor fibers contained in such surfaces. First, we prove that there is no irreducible surface of bidegree (1,d) containing d+2 twistor fibers in general position. On the other hand, given any collection of (d+1) twistor fibers satisfying a mild natural constraint, we prove the existence of a surface of bidegree (1,d) that contains them. We improve our results for d=2 or d=3, by removing all the generality hypotheses.

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arXiv2023

Weyl cycles on the blow-up of $\mathbb{P}^4$ at eight points

We define the Weyl cycles on $X^n_s$, the blown up projective space $\mathbb{P}^n$ in $s$ points in general position. In particular, we focus on the Mori Dream spaces $X^3_7$ and $X^{4}_{8}$, where we classify all the Weyl cycles of codimension two. We further introduce the Weyl expected dimension for the space of the global sections of any effective divisor that generalizes the linear expected dimension and the secant expected dimension.

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arXiv2022

Twistor geometry of the Flag manifold

A study is made of algebraic curves and surfaces in the flag manifold $\mathbb{F}=SU(3)/T^2$, and their configuration relative to the twistor projection $π$ from $\mathbb{F}$ to the complex projective plane $\mathbb{CP}^2$, defined with the help of an anti-holomorphic involution $j$. This is motivated by analogous studies of algebraic surfaces of low degree in the twistor space $\mathbb{CP}^3$ of $S^4$. Deformations of twistor fibres project to real surfaces in $\mathbb{CP}^2$, whose metric geometry is investigated. Attention is then focussed on toric Del Pezzo surfaces that are the simplest type of surfaces in $\mathbb{F}$ of bidegree $(1,1)$. These surfaces define orthogonal complex structures on specified dense open subsets of $\mathbb{CP}^2$ relative to its Fubini-Study metric. The discriminant loci of various surfaces of bidegree $(1,1)$ are determined, and bounds given on the number of twistor fibres that are contained in more general algebraic surfaces in $\mathbb{F}$.

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arXiv2022

Surfaces in the flag threefold containing smooth conics and twistor fibers

We study smooth integral curves of bidegree $(1,1)$, called \textit{smooth conics}, in the flag threefold $\mathbb{F}$. The study is motivated by the fact that the family of smooth conics contains the set of fibers of the twistor projection $\mathbb{F}\to\mathbb{CP}^{2}$. We give a bound on the maximum number of smooth conics contained in a smooth surface $S\subset\mathbb{F}$. Then, we show qualitative properties of algebraic surfaces containing a prescribed number of smooth conics. Lastly, we study surfaces containing infinitely many twistor fibers. We show that the only smooth cases are surfaces of bidegree $(1,1)$. Then, for any integer $a>1$, we exhibit a method to construct an integral surface of bidegree $(a,a)$ containing infinitely many twistor fibers.

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arXiv2015

On the effective cone of $\mathbb{P}^n$ blown-up at $n+3$ points

We compute the facets of the effective and movable cones of divisors on the blow-up of $\mathbb{P}^n$ at $n+3$ points in general position. Given any linear system of hypersurfaces of $\mathbb{P}^n$ based at $n+3$ multiple points in general position, we prove that the secant varieties to the rational normal curve of degree $n$ passing through the points, as well as their joins with linear subspaces spanned by some of the points, are cycles of the base locus and we compute their multiplicity. We conjecture that a linear system with $n+3$ points is linearly special only if it contains such subvarieties in the base locus and we give a new formula for the expected dimension.

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arXiv2015

On linear systems of $\mathbb{P}^3$ with nine base points

We study special linear systems of surfaces of $\mathbb{P}^3$ interpolating nine points in general position having a quadric as fixed component. By performing degenerations in the blown-up space, we interpret the quadric obstruction in terms of linear obstructions for a quasi-homogeneous class. By degeneration we also prove a Nagata type result for $\mathbb{P}^2$ that implies a base locus lemma for the quadric. As an application we establish Laface-Ugaglia Conjecture for linear systems with multiplicities bounded by 8 and for homogeneous linear systems with multiplicity m and degree up to 2m+1.

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arXiv2013

On a notion of speciality of linear systems in P^n

Given a linear system in P^n with assigned multiple general points we compute the cohomology groups of its strict transforms via the blow-up of its linear base locus. This leads us to give a new definition of expected dimension of a linear system, which takes into account the contribution of the linear base locus, and thus to introduce the notion of linear speciality. We investigate such a notion giving sufficient conditions for a linear system to be linearly non-special for arbitrary number of points, and necessary conditions for small numbers of points.

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arXiv2012

Secant varieties of Segre-Veronese varieties P^m x P^n embedded by O(1,2)

Let $X_{m,n}$ be the Segre-Veronese variety $\mathbb{P}^m \times \mathbb{P}^n$ embedded by the morphism given by $\mathcal{O}(1,2)$. In this paper, we provide two functions $\underline{s}(m,n)\le \bar{s}(m,n)$ such that the $s^{\mathrm{th}}$ secant variety of $X_{m,n}$ has the expected dimension if $s \leq \underline{s}(m,n)$ or $ \bar{s}(m,n) \leq s$. We also present a conjecturally complete list of defective secant varieties of such Segre-Veronese varieties.

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