相关论文: The Ciliberto-Di Gennaro conjecture for $d=5$

The Ciliberto-Di Gennaro conjecture addresses the factoriality of three-dimensional nodal hypersurfaces, and their geometric properties. We prove this conjecture for hypersurfaces of degree 6 by adapting a recent technique due to R.…

In this paper we prove that a nodal hypersurface in P^4 with defect has at least (d-1)^2 nodes, and if it has at most 2(d-2)(d-1) nodes and d>6 then it contains either a plane or a quadric surface. Furthermore, we prove that a nodal double…

We prove the factoriality of a nodal hypersurface in $\mathbb{P}^{4}$ of degree $d$ that has at most $2(d-1)^{2}/3$ singular points, and factoriality of a double cover of $\mathbb{P}^{3}$ branched over a nodal surface of degree $2r$ having…

We prove that for n= 5, 6, 7, a nodal hypersurface of degree n in P^4 is factorial if it has at most (n-1)^2-1 nodes.

J.-P. Roudneff conjectured in 1991 that every arrangement of $n \ge 2d+1\ge 5$ pseudohyperplanes in the real projective space $\mathbb{P}^d$ has at most $\sum_{i=0}^{d-2} \binom{n-1}{i}$ complete cells (i.e., cells bounded by each…

In this text we show that the deformation space of a nodal surface $X$ of degree $d$ is smooth and of the expected dimension if $d\leq 7$ or $d\geq 8$ and $X$ has at most $4d-5$ nodes. (The case $d\leq 7$ was previously covered by Alexandru…

Let $X$ be an arbitrary smooth hypersurface in $\mathbb{C} \mathbb{P}^n$ of degree $d$. We prove the de Jong-Debarre Conjecture for $n \geq 2d-4$: the space of lines in $X$ has dimension $2n-d-3$. We also prove an analogous result for…

We show the existence of surfaces of degree $d$ in $\dP^3(\dC)$ with approximately ${3j+2\over 6j(j+1)} d^3$ singularities of type $A_j, 2\le j\le d-1$. The result is based on Chmutov's construction of nodal surfaces. For the proof we use…

Finding a good bound on the maximal edge diameter $\Delta(d,n)$ of a polytope in terms of its dimension $d$ and the number of its facets $n$ is one of the basic open questions in polytope theory \cite{BG}. Although some bounds are known,…

In projective dimension growth results, one bounds the number of rational points of height at most $H$ on an irreducible hypersurface in $\mathbb P^n$ of degree $d>3$ by $C(n)d^2 H^{n-1}(\log H)^{M(n)}$, where the quadratic dependence in…

According to the G\"ottsche conjecture (now a theorem), the degree N^{d, delta} of the Severi variety of plane curves of degree d with delta nodes is given by a polynomial in d, provided d is large enough. These "node polynomials"…

Let $k$ and $d$ be such that $k \ge d+2$. Consider two $k$-colourings of a $d$-degenerate graph $G$. Can we transform one into the other by recolouring one vertex at each step while maintaining a proper coloring at any step? Cereceda et al.…

The Nagata Conjecture is one of the most intriguing open problems in the area of curves in the plane. It is easily stated. Namely, it predicts that the smallest degree d of a plane curve passing through r $\ge$ 10 general points in the…

The result of Padrol asserts that for every $d\geq 4$, there exist $2^{\Omega(n\log n)}$ distinct combinatorial types of $\lfloor d/2\rfloor$-neighborly simplicial $(d-1)$-spheres with $n$ vertices. We present a construction showing that…

We prove the $\mathbb{Q}$-factoriality of a nodal hypersurface in $\mathbb{P}^{4}$ of degree $n$ with at most ${\frac{(n-1)^{2}}{4}}$ nodes and the $\mathbb{Q}$-factoriality of a double cover of $\mathbb{P}^{3}$ branched over a nodal…

Chisini's conjecture asserts that for a cuspidal curve $B\subset \mathbb P^2$ a generic morphism $f$ of a smooth projective surface onto $\mathbb P^2$ of degree $\geq 5$, branched along $B$, is unique up to isomorphism. We prove that if…

Let $X \subset \mathbb{P}^{n+1}$ be a smooth Fano hypersurface of dimension $n$ and degree $d$. The derived category of coherent sheaves on $X$ contains an interesting subcategory called the Kuznetsov component $\mathcal{A}_X$. We show that…

For $d$ ranging from 2 to 6, we prove that the web by conics naturally defined on any smooth del Pezzo surface of degree $d$ carries an interesting functional identity whose components all are a certain antisymmetric hyperlogarithm of…

This is an expository paper based on the results in [12] and [16]. The main goal is to prove the following two conjectures for genus up to two. (1) Witten's conjecture on the relations between higher spin curves and Gelfand-Dickey…

Let $A,B \subseteq \mathbb{R}^d $ both span $\mathbb{R}^d$ such that $\langle a, b \rangle \in \{0,1\}$ holds for all $a \in A$, $b \in B$. We show that $ |A| \cdot |B| \le (d+1) 2^d $. This allows us to settle a conjecture by Bohn, Faenza,…