Pietro De Poi, Francesco Zucconi
We extend the classical Enriques-Petri Theorem to $s$-subcanonical projectively normal curves, proving that such a curve is $(s+2)$-gonal if and only if it is contained in a surface of minimal degree. Moreover, we show that any Fermat hypersurface of degree $s+2$ is apolar to an $s$-subcanonical $(s+2)$-gonal projectively normal curve, and vice versa.
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Let $X$ be a codimension 1 subvariety of dimension $>1$ of a variety of minimal degree $Y$. If $X$ is subcanonical with Gorenstein canonical singularities admitting a crepant resolution, then $X$ is Arithmetically Gorenstein and we characterise such subvarieties $X$ of $Y$ via apolarity as those whose apolar hypersurfaces are Fermat.
Limit F-invariants of the simple singularities and the Fermat hypersurfaces
本文计算了简单奇点与费马超曲面的极限F-不变量，推广了相关公式，并将正则环刻画与Watanabe--Yoshida猜想推广至特征零。
Gherardelli linkage and complete intersections
我们的主要定理刻画了特征为0的代数闭域上的3维或更高维射影空间中余维2的完全交为次正则和自链接子概型。为了证明这个定理，我们将证明盖拉尔代利链接定理，它断言两个超曲面的部分相交是次正则的当且仅当它的剩余相交是，在方案理论上，两个超曲面与第三个超曲面的相交。
We study the critical points of the likelihood function over the Fermat hypersurface. This problem is related to one of the main problems in statistical optimization: maximum likelihood estimation. The number of critical points over a projective variety is a topological invariant of the variety and is called maximum likelihood degree. We provide closed formulas for the maximum likelihood degree of any Fermat curve in the projective plane and of Fermat hypersurfaces of degree 2 in any projective space. Algorithmic methods to compute the ML degree of a generic Fermat hypersurface are developed t…
We investigate the hypersurfaces which are the generation of the Fermat hypersufaces, and determine their projective isomorphism classes.
This paper studies the existence of free and very free curves on the degree 5 Fermat hypersurface in P^5 over a field of characteristic 2. We find that such curves exist in degrees 8 and 9 and not in lower degrees.
The variety of all smooth hypersurfaces of given degree and dimension has the Fermat hypersurface as a natural base point. In order to study the period map for such varieties, we first determine the integral polarized Hodge structure of the primitive cohomology of a Fermat hypersurface (as a module over the automorphism group of the hypersurface). We then focus on the degree 3 case and show that the period map for cubic fourfolds as analyzed by R. Laza and the author gives complete information about the period map for cubic hypersurfaces of lower dimension dimension. In particular, we thus rec…
Hyperbolic hypersurfaces in P^n of Fermat-Waring type
本文证明了在P^n中存在4（n-1）^2次的双曲费马-华林型超曲面的代数族，其中n ≥ 1。此外，在P^n中存在次数为4n^2-2n+1的双曲Fermat-Waring超曲面，它们具有完全双曲的、双曲嵌入的补曲面。
On Smooth Divisors of a Projective Hypersurface
本文证明了P^n，n>4（特征为零的代数闭域上的射影空间）超曲面的光滑除子次数的一个有效界.我们的结果如下从一个强有力的（因为程度的除数是不参与）推广的“特殊性定理”的Gruson-Peskine，我们证明持有任何顺利，subcanonical，余维2，投影verieties维至少3。
On real forms of Fermat hypersurfaces
本文计算了次数 $d \ge 2$ 的 Fermat 超曲面（次数4曲面除外）的实形式数量，并给出了它们的显式描述。
In this note we study rational curves on degree $p^r+1$ Fermat hypersurface in $\PP^{p^r+1}_k$, where $k$ is an algebraically closed field of characteristic $p$. The key point is that the presence of Frobenius morphism makes the behavior of rational curves to be very different from that of charateristic 0. We show that if there exists $N_0$ such that for all $e\geq N_0$ there is a degree $e$ very free rational curve on $X$, then $N_0> p^r(p^r-1)$.
We study the intersection of two particular Fermat hypersurfaces in $\mathbb{P}^3$ over a finite field. Using the Kani-Rosen decomposition we study arithmetic properties of this curve in terms of its quotients. Explicit computation of the quotients is done using a Gröbner basis algorithm. We also study the $p$-rank, zeta function, and number of rational points, of the modulo $p$ reduction of the curve. We show that the Jacobian of the genus 2 quotient is $(4,4)$-split.
Finding the point in an algebraic variety that is closest to a given point is an optimization problem with many applications. We study the case when the variety is a Fermat hypersurface. Our formula for its Euclidean distance degree is a piecewise polynomial whose pieces are defined by subtle congruence conditons.
We produce special Lagrangian $T^n$-fibrations on the generic regions of some Calabi-Yau hypersurfaces in the Fermat family $X_s=\{ Z_0\ldots Z_{n+1}+ e^{-s} ( Z_0^{n+2}+ \ldots Z_{n+1}^{n+2} ) =0 \}\subset \mathbb{CP}^{n+1} $ near the large complex structure limit $s\to +\infty$.
Maximum likelihood degree of a projective variety is the number of critical points of a general likelihood function. In this note, we compute the Maximum likelihood degree of Fermat hypersurfaces. We give a formula of the Maximum likelihood degree in terms of the constants $β_{μ, ν}$, which is defined to be the number of complex solutions to the system of equations $z_1^ν=z_2^ν=\cdots=z_μ^ν=1$ and $z_1+\cdots +z_μ+1=0$.
The orbifold Hochschild product for Fermat hypersurface
本文研究费马超曲面轨形上多向量场上同调的积结构与双分次，证明其积具有结合性，并验证了 Calabi-Yau 情形下与 Hochschild 上同调及同调镜像对称的预期一致。
Lagrangian torus fibration of quintic Calabi-Yau hypersurfaces I: Fermat quintic case
本文利用梯度流方法构造了Fermat型五次\cy超曲面的Lagrange环面纤维化。我们还计算了期望的特殊拉格朗日环面纤维化的单值性，并讨论了奇异纤维的结构。
Level structures on cyclic covers of $\mathbb{P}^n$ and the homology of Fermat hypersurfaces
本文研究了循环覆盖上的水平结构与费马超曲面的同调关系，通过三次超曲面的例子回答了Beauville的问题。
The zeta-function of monomial deformations of Fermat hypersurfaces
本文试图对最近在镜像对称的背景下得到的关于簇族的ζ函数的结果给出一个数学解释。在这样做的时候，我们获得了具体和明确的例子，最近使用的算法中的一些结果，光滑超曲面P^n上的点计数。特别是，我们扩展了Kadir和Yui的单名动机对应，并给出了与Fermat超曲面的单名变形相关的p-adic Picard-Fuchs方程的显式解。作为一个副产品，我们得到了Poincare对偶的刚性上同调的某些奇异仿射簇。
Let $\mathbb{F}_q$ be a finite field with $q=p^n$ elements. In this paper, we study the number of $\mathbb{F}_q$-rational points on the affine hypersurface $\mathcal X$ given by $a_1 x_1^{d_1}+\dots+a_s x_s^{d_s}=b$, where $b\in\mathbb{F}_q^*$. A classic well-konwn result from Weil yields a bound for such number of points. This paper presents necessary and sufficient conditions for the maximality and minimality of $\mathcal X$ with respect to Weil's bound.
Degeneracy of holomorphic mappings into or avoiding Fermat type hypersurfaces
本文证明了进入 Fermat 型超曲面的全纯映射的退化性，强化了 Green 的经典结论，并为 Etesse 的近期结果提供了 Nevanlinna 理论证明。
Twisted forms of classical hypersurfaces
该研究在实数域和有限域上计数费马、德尔萨特、克莱因超曲面的扭曲形式，推广微分方法到正特征域，计算其正特征自同构群，重现了费马超曲面实形式的相关定理。
We characterize the smallest codimension components of the Hodge locus of smooth degree $d$ hypersurfaces of the projective space $\mathbb{P}^{n+1}$ of even dimension $n$, passing through the Fermat variety (with $d\neq 3,4,6$). They correspond to the locus of hypersurfaces containing a linear algebraic cycle of dimension $\frac{n}{2}$. Furthermore, we prove that among all the local Hodge loci associated to a non-linear cycle passing through Fermat, the ones associated to a complete intersection cycle of type $(1,1,\ldots,1,2)$ attain the minimal possible codimension of their Zariski tangent s…
Permutation-equivariant quantum K-theory of Fermat singularities
本文计算了费马奇点的零 genus 置换对称量子K理论，扩展了Givental-Tonita的adelic拉格朗日锥形式，并获得满足相同q-差分方程的I函数，扩展了Landau-Ginzburg/Calabi-Yau对应关系，但在K理论中出现差异。
The most fundamental example of mirror symmetry compares the Fermat hypersurfaces in P^n and P^n/G, where G is a finite group that acts on P^n and preserves the Fermat hypersurface. We generalise this to hypersurfaces in Grassmannians, where the picture is richer and more complex. There is a finite group G that acts on the Grassmannian Gr(n,r) and preserves an appropriate Calabi-Yau hypersurface. We establish how mirror symmetry, toric degenerations, blow-ups and variation of GIT relate the Calabi-Yau hypersurfaces inside Gr(n,r) and Gr(n,r)/G. This allows us to describe a compactification of …
Retraction of the complement of smooth projective hypersurfaces to an $n$-dimensional $Δ$-complex
本文通过分支覆盖结构，将Fermat曲线补集显式构造为二维Δ-复形的强形变收缩核。
Free curves in Fano hypersurfaces must have high degree
本文证明在正特征域中，Fano超曲面中自由有理曲线的最小次数无法由n的线性函数界定，并给出了超线性的界。
GKZ-Generalized Hypergeometric Systems in Mirror Symmetry of Calabi-Yau Hypersurfaces
本文详细研究了由Gel'fand，Kapranov和Zelevinski在复曲面几何中引入的广义超几何系统（GKZ-超几何系统）。GKZ系统是在Calabi-Yau复曲面簇的模理论中自然产生的，在镜像对称性的应用中起着重要的作用。我们发现，所谓的复曲面理想的Gröbner基础确定了GKZ系统的局部解的微分算子的有限集。在称为大半径极限的特殊点处，我们发现GKZ系统中算子的主部与复曲面簇的交环有密切的关系。作为应用，我们分析了Hodge数达到$h^{1,1}=3$的一般三维Fermat型和非Fermat型超曲面.我们还发现并分析了几个与奇异模型相关的非Landau-Ginzburg模型。
Smooth Divisors of Projective Hypersurfaces
我们研究了任意特征的代数闭域。Ellingsrud-Peskine证明了P^4中的光滑曲面受到很强的限制。他们的整个论点是来自这样一个事实，即部分亏格的表面程度d躺在超曲面的程度s变化的间隔长度\frac{d(s-1)^2}{2s}。本文的目的是证明对于P^n的光滑余维两个子簇，n \geq 5，当区间的长度仅取决于s时，可以得到类似的结果.本文的重点是引理1.1，它的证明是N_X（-1）的正性的直接应用（其中N_X是X在P^n中的正规丛）。我们得到一系列（n-3）不等式;其中第一个是在文件Ellingsrud-Peskine，第二个是在一个初步的版本（arXiv：数学。AG/0406497）由一个基本上等价的，但更多的几何参数。首先，我们可以得出两个结论：（1）粗略地说，（Thm。2.1，注2.2）位于s次超曲面上的P^5的光滑子簇的“biliaison类”族是有限的; 2）位于s次超曲面上的P^6的光滑余维2子簇的族是有限的（Thm. 1.4）。1）中引用的结果是无效的，但2）是有效的。在最后一节中，我们试图得到精确的不等式，这些不等式将P^n，n \geq 5的光滑次正则子簇X的通常数值不变量（包含X的超曲面的次数d，特徴指数e，最小次数s）联系起来。特别是，我们证明（Thm。3.12）：s \geq n+1。
Traditional algebraic geometric invariants lose some of their potency in positive characteristic. For instance, smooth projective hypersurfaces may be covered by lines despite being of arbitrarily high degree. The purpose of this dissertation is to define a class of hypersurfaces that exhibits such classically unexpected properties, and to offer a perspective with which to conceptualize such phenomena. Specifically, this dissertation proposes an analogy between the eponymous $q$-bic hypersurfaces -- special hypersurfaces of degree $q+1$, with $q$ any power of the ground field characteristic, a…
The Eckardt hypersurface in $\mathbb{P}^{19}$ parameterizes smooth cubic surfaces with an Eckardt point, which is a point common to three of the $27$ lines on a smooth cubic surface. We describe the cubic surfaces lying on the singular locus of the model of this hypersurface in $\mathbb{P}^4$, obtained via restriction to the space of cubic surfaces possessing a so-called Sylvester form. We prove that inside the moduli of cubics, the singular locus corresponds to a reducible surface with two rational irreducible components intersecting along two rational curves. The two curves intersect in two …
Automorphism groups of rigid complete intersections
研究射影空间中次数严格递增的超曲面完全交的自同构群，利用组合刚性条件证明其自同构群是定义超曲面自同构群的交集，并应用此原理于两个超曲面完全交的自然族，确定了相关自同构群。
Projective Lifting Theorem, Complex Reflection Groups, and Quasi-Galois Points for Fermat Varieties
本文证明保持点投影的超曲面双有理自同构可唯一提升为射影变换，并利用复反射群理论确定了Fermat簇的Galois点与拟Galois点的精确数量。
Let $f$ be a function on a finite field $F$. The decomposition of the generalized Fermat variety $X$ defined by the multivariate polynomial of degree $n$, $ϕ(x,y,z)=f(x)+f(y)+f(z)$ in $\Bbb{P}^3(\overline{\mathbb{F}}_2)$, plays a crucial role in the study of almost perfect non-linear (APN) functions and exceptional APN functions. Their structure depends fundamentally on the Fermat varieties corresponding to the monomial functions of exceptional degrees $n=2^k+1$ and $n=2^{2k}-2^k+1$ (Gold and Kasami-Welch numbers, respectively). Very important results for these have been obtained by Janwa, McG…
We study Galois points for a hypersurface $X$ with $\dim {\rm Sing}(X) \le \dim X-2$. The purpose of this article is to determine the set $Δ(X)$ of Galois points in characteristic zero: Indeed, we give a sharp upper bound of the number of Galois points in terms of $\dim X$ and $\dim {\rm Sing}(X)$ if $Δ(X)$ is a finite set, and prove that $X$ is a cone if $Δ(X)$ is infinite. To achieve our purpose, we need a certain hyperplane section theorem on Galois point. We prove this theorem in arbitrary characteristic. On the other hand, the hyperplane section theorem has other important applications: F…
On fake linear cycles inside Fermat varieties
本文引入并刻画了费马簇中的假线性闭链，证明其仅在次数为3、4、6时存在且无穷多，并利用伽罗瓦作用与IVHS二阶不变量揭示了其病态几何性质，推广了关于霍奇轨迹最小余维数分支的已有结果。
A Dwork family is a one-parameter monomial deformation of a Fermat hypersurface. In this paper we compute algebraically the invariant part of its Gauss-Manin cohomology under the action of certain subgroup of automorphisms. To achieve that goal we use the algebraic theory of $\mathcal{D}$-modules, especially one-dimensional hypergeometric ones.
Higher dimensional examples of manifolds whose adjoint bundles are not spanned
令$(X,L)$是$n$维极化簇。Fujita的猜想说，如果$L^n>1$，那么伴随丛$K_X+nL$是张成的，并且$K_X+(n+1)L$是非常充足的。有一些例子，$K_X+nL$不是跨越或$K_X+(n+1)L$不是很充足。这些是$(¶^n,Ø(1))$，超曲面$M$的程度$6$在加权射影空间$¶(3,2,1,1,\cdots ,1)$与$Ø_M(1)$和数值Godeaux表面等。数值Godeaux表面是商空间的费马型超曲面的程度$5$在$¶^3$的作用顺序$5$。这些例子并不多。我们为任何维度构建新的示例。
MSP theory for smooth Calabi-Yau threefolds in weighted $\mathbb{P}^4$
本文将五次三维流形的 MSP 理论推广到三类加权射影空间中的 Fermat 型 Calabi-Yau 三维流形，建立相应的 $N$-混合自旋-$P$ 场理论。
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