Factorial hypersurfaces in P^4 with nodes

On factoriality of nodal threefolds

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 surface of degree $2r$ with at most ${\frac{(2r-1)r}{3}}$ nodes.

Factorial threefolds and Shokurov vanishing

We prove the factoriality of the following nodal threefolds: a complete intersection of hypersurfaces $F$ and $G\subset\mathbb{P}^{5}$ of degree $n$ and $k$ respectively, where $G$ is smooth, $|\mathrm{Sing}(F\cap G)|\leqslant(n+k-2)(n-1)/5$, $n\geqslant k$; a double cover of a smooth hypersurface $F\subset\mathbb{P}^{4}$ of degree $n$ branched over a surface that is cut out on $F$ by a hypersurface $G$ of degree $2r\geqslant n$, and $|\mathrm{Sing}(F\cap G)|\leqslant(2r+n-2)r/4$.

Factoriality of complete intersection threefolds

Let X be a complete intersection of two hypersurfaces F_n and F_k in the projective space P^5 of degree n and k respectively with n >= k, such that the singularities of X are nodal and F_k is smooth. We prove that if the threefold X has at most (n+k-2)(n-1)-1 singular points, then it is factorial.

On the locus of non-rigid hypersurfaces

We show that the Zariski closure of the set of hypersurfaces of degree $M$ in ${\mathbb P}^{M}$, where $M\geq 5$, which are either not factorial or not birationally superrigid, is of codimension at least $\binom{M-3}{2}+1$ in the parameter space.

Hypersurfaces with defect

A projective hypersurface $X \subseteq \mathbb P^n$ has defect if $h^i(X) \neq h^i(\mathbb P^n)$ for some $i \in \{n, \dots, 2n-2\}$ in a suitable cohomology theory. This occurs for example when $X \subseteq \mathbb P^4$ is not $\mathbb Q$-factorial. We show that in characteristic 0, the Tjurina number of hypersurfaces with defect is large. For $X$ with mild singularities, there is a similar result in positive characteristic. As an application, we obtain a lower bound on the asymptotic density of hypersurfaces without defect over a finite field.

论极大非阶乘节点 Fano 三维簇

On maximally non-factorial nodal Fano threefolds

本文研究非阶乘节点 Fano 三维簇，完成了对恰含 1 个节点且类群秩为 2 情形的分类。

Non-rational nodal quartic threefolds

The $\mathbb{Q}$-factoriality of a nodal quartic 3-fold implies its non-rationality. We prove that a nodal quartic 3-fold with at most 8 nodes is $\mathbb{Q}$-factorial, and we show that a nodal quartic 3-fold with 9 nodes is not $\mathbb{Q}$-factorial if and only if it contains a plane. However, there are non-rational non-$\mathbb{Q}$-factorial nodal quartic 3-folds in $\mathbb{P}^4$. In particular, we prove the non-rationality of a general non-$\mathbb{Q}$-factorial nodal quartic 3-fold that contains either a plane or a smooth del Pezzo surface of degree 4.

A note on octic hypersurfaces with many nodes

In the present note we use rank-2-bundles over ${\bb P}^3$ to construct octic hypersurfaces with many nodes. We give an example with 128 nodes.

A Quintic Hypersurface in $\PP^8(\CC)$ with Many Nodes

We construct a hypersurface of degree 5 in projective space $\PP^8(\CC)$ which contains exactly 23436 ordinary nodes and no further singularities. This limits the maximum number $μ_{8}(5)$ of ordinary nodes a hyperquintic in $\PP^8(\CC)$ can have to $23436 \leq μ_{8}(5) \leq 27876$. Our method generalizes the approach by the $3^{\text{rd}}$ author for the construction of a quintic threefold with 130 nodes in an earlier paper.

Points in projective spaces and applications

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 less than $(2r-1)r$ singular points.

Enumeration of algebraic and tropical singular hypersurfaces

We develop a version of Mikhalkin's lattice path algorithm for projective hypersurfaces of arbitrary degree and dimension, which enumerates singular tropical hypersurfaces passing through appropriate configuration of points. By proving a correspondence theorem combined with the lattice path algorithm, we construct a $δ$ dimensional linear space of degree $ d $ real hypersurfaces containing $\frac{1}{δ!} (γ_nd^n)^δ+ O(d^{nδ-1})$ hypersurfaces with $ δ$ real nodes, where $ γ_n $ are positive and given by a recursive formula. This is asymptotically comparable to the number $ \frac{1}{δ!} \left( (…

从非常一般的超曲面的映射

Morphisms from a very general hypersurface

研究非常一般超曲面的映射性质，证明在特定条件下Y是klt Fano变种，并给出映射度数的上界。

节点曲面的簇、编码理论与三次超曲面的判别式。第一部分：一般理论与节点K3曲面。第二部分：三次超曲面及其关联判别式。第三部分：节点五次曲面。第四部分：节点六次曲面

Varieties of Nodal surfaces, coding theory and Discriminants of cubic hypersurfaces. Part 1: Generalities and nodal K3 surfaces. Part 2: Cubic Hypersurfaces, associated discriminants. Part 3: Nodal quintics. Part 4: Nodal sextics

本文通过为射影节点曲面附加二元码，系统研究了P^3中节点曲面的Severi簇的不可约分量及其关联层级，完整分类了节点K3曲面，并揭示了Togliatti五次曲面及最大节点数六次曲面与三次超曲面判别式之间的深刻联系。

具有普通 $m$ 重点的超曲面的节点形变

Nodal deformations of hypersurfaces with an ordinary $m$-fold point

本文研究具有普通 $m$ 重点的节点超曲面的形变，利用谱半连续性给出节点数上界，并构造多项式下界，改进了 Ciliberto-Galati 的结果。

锥形消解与节点超曲面模空间的上同调

Conical resolutions and cohomology of the moduli spaces of nodal hypersurfaces

本文利用锥形消解及其推广形式，并结合紧 GIT 商的显式描述，计算节点三次、四次平面曲线及三次曲面方程空间与模空间的有理上同调。

d=5时的Ciliberto-Di Gennaro猜想

The Ciliberto-Di Gennaro conjecture for $d=5$

本文证明了d=5时的Ciliberto-Di Gennaro猜想，填补了该猜想在d≥3情况下的空白。

具有亏格的节点簇的极大族

Maximal families of nodal varieties with defect

本文研究射影空间中具有亏格的节点超曲面，给出节点数的下界，刻画小节点数情形的几何结构，并构造了最大维数的节点超曲面族。

A Division Theorem for Nodal Projective Hypersurfaces

Let $V_{n,d}$ be the variety of equations for hypersurfaces of degree $d$ in $\mathbb{P}^n(\mathbb{C})$ with singularities not worse than simple nodes. We prove that the orbit map $G'=SL_{n+1}(\mathbb{C}) \to V_{n,d}$, $g\mapsto g\cdot s_0$, $s_0\in V_{n,d}$ is surjective on the rational cohomology if $n>1$, $d\geq 3$, and $(n,d)\neq (2,3)$. As a result, the Leray-Serre spectral sequence of the map from $V_{n,d}$ to the homotopy quotient $(V_{n,d})_{hG'}$ degenerates at $E_2$, and so does the Leray spectral sequence of the quotient map $V_{n,d}\to V_{n,d}/G'$ provided the geometric quotient $V…

On Wahl's proof of $μ(6)=65$

D. Jaffe and D. Ruberman proved in 1997 that a sextic hypersurface in $\mathbb{P}^3$ has at most 65 nodes (the bound is sharp by Barth's construction). Almost at the same time, J. Wahl proposed a much shorter proof of the same result, by proving that a linear code $V\subset \F^{66}$ with weights in $\{24,32,40\}$ has dimension $\dim(V)\leq12$. He claimed that Jaffe-Ruberman's theorem follows as a corollary since the code associated to a sextic with n nodes has dimension at least $n-53$ and an incorrect result stated by Casnati and Catanese asserted that the possible cardinalities of an even se…

The Castelnuovo-Mumford regularity of an integral variety of a vector field on projective space

The Castelnuovo-Mumford regularity r of a complex, projective variety V is an upper bound for the degrees of the hypersurfaces necessary to cut out V. In this note we give a bound for r when V is left invariant by a vector field on the ambient projective space. More precisely, assume V is arithmetically Cohen-Macaulay, for instance, a complete intersection. Assume as well that V projects to a normal-crossings hypersurface, which is the case when V is a curve with at most ordinary nodes. Then we show that r<m+s+2, where s is the dimension of V and m is the degree of the vector field. Our method…

The class of Eisenbud--Khimshiashvili--Levine is the local A1-Brouwer degree

Given a polynomial function with an isolated zero at the origin, we prove that the local A1-Brouwer degree equals the Eisenbud-Khimshiashvili-Levine class. This answers a question posed by David Eisenbud in 1978. We give an application to counting nodes together with associated arithmetic information by enriching Milnor's equality between the local degree of the gradient and the number of nodes into which a hypersurface singularity degenerates to an equality in the Grothendieck-Witt group.

$\mathbb{P}^3$ 中的节点曲面与编码理论

Nodal surfaces in $\mathbb{P}^3$ and coding theory

本文证明 $\mathbb{P}^3$ 中具有 65 个节点的六次曲面（如 Barth 曲面）关联的二元线性码唯一，并给出可能达到节点数上界的七次曲面的候选码。