Justin Sawon
Let $X\to¶^n$ be an irreducible holomorphic symplectic manifold of dimension $2n$ fibred over $¶^n$. Matsushita proved that the generic fibre is a holomorphic Lagrangian abelian variety. In this article we study the discriminant locus $Δ\subset¶^n$ parametrizing singular fibres. Our main result is a formula for the degree of $Δ$, leading to bounds on the degree when $X$ is a four-fold.
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In this paper we give the Weierstrass equations and the generators of Mordell-Weil groups for Jacobian fibrations on the singular K3 surface of discriminant 3.
We give a Kodaira-type classification of general singular fibers of a holomorphic Lagrangian fibration in Fujiki's class $\mathcal C$. Our approach is based on the study of the characteristic vector field of the discriminantal hypersurface, which naturally arises from the defining equation of the hypersurface via the symplectic form. As an application, we show that the characteristic foliation of the discriminantal hypersurface has algebraic leaves which are either rational curves or smooth elliptic curves.
Let $alb_X: X \rightarrow A$ be the Albanese map of a smooth projective variety and $f: X \rightarrow Y$ the fibration from the Stein factorization of $alb_X$. For a positive integer $m$, if $f$ and $m$ satisfy the assumptions AS(1,2), then the translates through the origin of all components of cohomological locus $V^0(ω_X^m, alb_X)$ generates $I^*Pic^0(S)$ where $I: X \rightarrow S$ denotes the Iitaka fibration. This result applies to studying pluricanonical maps. We also considered the problem about whether a fibration is isotrivial and isogenous to a product.
On the discriminant locus of a generic projection
研究一般投影的判别位势的几何性质，证明其与对偶簇的线性截面的对偶关系，并推导判别位势的纯度陈述。
Even if there are too many elliptic fibrations to investigate and describe on the singular $K3$ surface $Y_{10}$ of discriminant 72 and belonging to the Apéry-Fermi pencil $(Y_k)$, we find on it many interesting properties. For example some of its elliptic fibrations with 3-torsion section induce by 3-isogeny either an elliptic fibration of $Y_2$, the unique $K3$ surface of discriminant 8, or an elliptic fibration of other $K3$ surfaces of discriminant 72.
To a dominant morphism $π:X/S \to Y/S$ of Nœtherian integral $S$-schemes one has the inclusion $C_π\subset B_π$ of the critical locus in the branch locus of $B_π$. Conditions on the relative differentials $Ω_{X/Y}$, $Ω_{X/S}$, and $Ω_{Y/S}$ are stated that imply bounds on the codimensions of $ C_π$ and $ B_π$. We also give bounds on the codimension of the discriminant locus, proving and generalising a conjecture of I. Dolgachev.
We define local fibration structures for real map germs with strictly positive dimensional discriminant: a local fibration structure over the complement of the discriminant, and a complete local fibration structure which includes the stratified discriminant into the picture. We provide new classes of map germs endowed with such local fibration structures.
We study the symplectic geometry of derived intersections of Lagrangian morphisms. In particular, we show that for a functional $f : X \rightarrow \mathbb{A}_k^1$, the derived critical locus has a natural Lagrangian fibration $\textbf{Crit}(f) \rightarrow X$. In the case where $f$ is non-degenerate and the strict critical locus is smooth, we show that the Lagrangian fibration on the derived critical locus is determined by the Hessian quadratic form.
We exploit techniques from classical (real and complex) algebraic geometry for the study of the standard twistor fibration $π:\mathbb{CP}^{3}\to S^{4}$. We prove three results about the topology of the twistor discriminant locus of an algebraic surface in $\mathbb{CP}^{3}$. First of all we prove that, with the exception of two exceptional cases, the real dimension of the twistor discriminant locus of an algebraic surface is always equal to 2. Secondly we describe the possible intersections of a general surface with the family of twistor lines: we find that only 4 configurations are possible an…
We consider systems of Laurent polynomials with support on a fixed point configuration. In the non-defective case, the closure of the locus of coefficients giving a non-degenerate multiple root of the system is defined by a polynomial called the mixed discriminant. We define a related polynomial called the multivariate iterated discriminant, generalizing the classical Schäfli method for hyperdeterminants. This iterated discriminant is easier to compute and we prove that it is always divisible by the mixed discriminant. We show that tangent intersections can be computed via iteration if and onl…
We give a construction of Lagrangian torus fibrations with controlled discriminant locus on certain affine varieties. In particular, we apply our construction in the following ways. We find a Lagrangian torus fibration on the 3-fold negative vertex whose discriminant locus has codimension 2; this provides a local model for finding torus fibrations on compact Calabi-Yau 3-folds with codimension 2 discriminant locus. Then, we find a Lagrangian torus fibration on a neighbourhood of the one-dimensional stratum of a simple normal crossing divisor (satisfying certain conditions) such that the base o…
Hypergeometric Discriminants
本文引入与非常仿射局部完全交簇平坦族相关联的超几何系统，证明Euler判别轨迹为其奇异轨迹，提出超几何判别式概念并建立基于似然方程的公式。
The following divisors in the space Sym^{12} P^1 of twelve points on P^1 are actually the same: (A) the possible locus of the twelve nodal fibers in a rational elliptic fibration (i.e. a pencil of plane cubic curves); (B) degree 12 binary forms that can be expressed as a cube plus a square; (C) the locus of the twelve tangents to a smooth plane quartic from a general point of the plane; (D) the branch locus of a degree 4 map from a hyperelliptic genus 3 curve to P^1; (E) the branch locus of a degree 3 map from a genus 4 curve to P^1 induced by a theta-characteristic; and several more. The corr…
On the GKZ discriminant locus
本文证明GKZ判别 locus 是所有判别 locus 的直接并集，无需取闭包，回答了Kite和Segal的问题。
We prove fibration theorems à la Milnor for differentiable real maps with non isolated critical values. We study the situation for maps with linear discriminant, and prove that the concept of d-regularity is the key point for the existence of a Milnor fibration on the sphere. We also explain how one can modify the target space by homeomorphisms to linearize a general discriminant. Whenever the composed map is d-regular one has fibration on the sphere. Plenty of examples are discussed along the text.
Consider a space X with the singular locus of positive dimension, Z=Sing(X). Suppose both Z and X are locally complete intersections at each point. The transversal type of X along Z is generically constant but at some points of Z it degenerates. In the previous work we have introduced (locally) the discriminant of the transversal type, a subscheme of Z, that reflects these degenerations whenever the generic transversal type is "ordinary". We have established the basic local properties of the discriminant. In the current paper we consider the global case. We compute the equivalence class of the…
We bound the slope of sweeping curves in the fourgonal locus of the moduli space of genus g algebraic curves. Our results follow from some Bogomolov-type inequalities for weakly positive rank two vector bundles on ruled surfaces.
Parameterized systems of polynomial equations arise in many applications in science and engineering with the real solutions describing, for example, equilibria of a dynamical system, linkages satisfying design constraints, and scene reconstruction in computer vision. Since different parameter values can have a different number of real solutions, the parameter space is decomposed into regions whose boundary forms the real discriminant locus. This article views locating the real discriminant locus as a supervised classification problem in machine learning where the goal is to determine classific…
The 102581 flat toric elliptic fibrations over P^2 are identified among the Calabi-Yau hypersurfaces that arise from the 473800776 reflexive 4-dimensional polytopes. In order to analyze their elliptic fibration structure, we describe the precise relation between the lattice polytope and the elliptic fibration. The fiber-divisor-graph is introduced as a way to visualize the embedding of the Kodaira fibers in the ambient toric fiber. In particular in the case of non-split discriminant components, this description is far more accurate than previous studies. The discriminant locus and Kodaria fibe…
Let $(l,f):(C^2,0)\rightarrow (C^2,0)$ be the germ of a holomorphic mapping such that $l=0$ is a smooth curve which is not a branch of the singular curve $f=0$. The direct image of the critical locus of this mapping is called the discriminant curve. In this paper we study the pairs $(l,f)$ for which the discriminant curve is non-degenerate in the Kouchnirenko sense.
The singular Hitchin fibration, cameral data, and representation theory
本文研究具有任意约化结构群的Higgs丛模栈上的Hitchin纤维化，利用Higgs场的中心化子分析其奇异轨迹，通过阿贝尔化纤维化分解Hitchin映射，并推广Donagi-Gaitsgory的相机数据描述纤维，最后应用于实群并揭示与李代数表示论的联系。
We classify singular fibres over general points of the discriminant locus of projective complex Lagrangian fibrations on 4-dimensional holomorphic symplectic manifolds. The singular fibre F is the following either one: F is isomorphic to the product of an elliptic curve and a Kodaira singular fibre up to finite unramified covering or F is a normal crossing variety consisting of several copies of a minimal elliptic ruled surface of which the dual graph is Dynkin diagram of type A_n, $\tilde{A_n}$, $\tilde{D_n}$ or D_n.
We prove finiteness of hyperkaehler Lagrangian fibrations in any fixed dimension with fixed Fujiki constant and discriminant of the Beauville-Bogomolov-Fujiki lattice, up to deformation. We also prove finiteness of hyperkähler Lagrangian fibrations with an ample line bundle of a given degree on the general fiber of the fibration.
Elliptic fibrations and involutions on K3 surfaces
本文综述了带有非辛对合的 K3 曲面上椭圆纤维化的分类，在更一般的对合框架下重新审视了具有 2-元初 Néron--Severi 格的 K3 曲面的完整分类，并给出了有理椭圆曲面二次覆盖的 K3 曲面上极值椭圆纤维化的完整分类。
Finiteness of stable Lagrangian fibrations
该研究综述 hyperkähler Lagrangian 纤维化形变类的有限性，并证明给定判别除子下稳定 Lagrangian 纤维化的有限性。
The Néron model of a higher-dimensional Lagrangian fibration
本文研究高维射影拉格朗日纤维化，在 $X'\to B$ 满射条件下构造阿贝尔纤维化及其自同构阿贝尔概形的 Néron 模型，并分析其挠子结构。
This is an announcement of the following construction: given an integral affine manifold $B$ with singularities, we build a topological space $X$ which is a torus fibration over $B$. The main new feature of the fibration $X\to B$ is that it has the discriminant in codimension 2.
Let Y->P^n be a flat family of integral Gorenstein curves, such that the compactified relative Jacobian X=\bar{J}^d(Y/P^n) is a Lagrangian fibration. We prove that the degree of the discriminant locus Delta in P^n is at least 4n+2, and we prove that X is a Beauville-Mukai integrable system if the degree of Delta is greater than 4n+20.
Whitney fold and cusp for algebraic surfaces, and singularities of discriminants
本文研究代数曲面投影的Whitney型奇点稳定性，通过分析判别式奇点轨迹和Vandermonde/Schur矩阵退化，在温和组合条件下枚举所有稳定的多奇点类型。
On the unirationality of conic bundles with discriminant of degree eight
本文研究了判别次数为八的锥面的不可理化性，通过构造有理多截面和变形理论分析，揭示了其在不同域上的不可理化性质。
In his proof of the fundamental lemma, Ngô established the product formula for the Hitchin fibration over the anisotropic locus. One expects this formula over the larger generically regular semisimple locus, and we confirm this by deducing the relevant vanishing statement for torsors over loop groups $R((t))$ from a general formula for $\mathrm{Pic}(R((t)))$. In the build up to the product formula, we present general algebraization, approximation, and invariance under Henselian pairs results for torsors, give short new proofs for the Elkik approximation theorem and the Chevalley isomorphism $\…
Let $f\colon X\to Z$ be a Mori fibre space. McKernan conjectured that the singularities of $Z$ are bounded in terms of the singularities of $X$. Shokurov generalised this to pairs: let $(X,B)$ be a klt pair and $f\colon X\to Z$ a contraction such that $K_X+B\sim_\R 0/Z$ and that the general fibres of $f$ are Fano type varieties; adjunction for fibre spaces produces a discriminant divisor $B_Z$ and a moduli divisor $M_Z$ on $Z$. it is then conjectured that the singularities of $(Z,B_Z+M_Z)$ are bounded in terms of the singularities of $(X,B)$. We prove Shokurov conjecture when $(F,\Supp B_F)$ b…
We explore a number of examples of special Lagrangian fibrations on non-compact Calabi-Yau manifolds invariant under torus actions. These include fibrations on crepant resolutions of canonical toric singularities (already found by Goldstein), proper versions of these fibrations, and fibrations on flat deformations of canonical toric singularities. We do this with an eye towards understanding the global structure and discriminant loci of such fibrations. The paper ends with some speculation, both about local mirror symmetry, and the connections of this work with the work of W.D. Ruan and D. Joy…
In this paper we give a construction of Lagrangian torus fibration for Calabi-Yau hypersurface in toric variety via the method of gradient flow. Using our construction of Lagrangian torus fibration, we are able to prove the symplectic topological version of SYZ mirror conjecture for generic Calabi-Yau hypersurface in toric variety. We will also be able to give precise formulation of SYZ mirror conjecture in general (including singular locus and duality of singular fibres).
Let $(X,L,V)$ be a triplet where $X$ is an irreducible smooth complex projective variety, $L$ is an ample and spanned line bundle on $X$ and $V\subseteq H^0(X,L)$ spans $L$. The discriminant locus $\Cal D(X,V) \subset |V|$ is the algebraic subset of singular elements of $|V|$. We study the components of $\Cal D(X,V)$ in connection with the jumping sets of $(X,V)$, generalizing the classical biduality theorem. We also deal with the degree of the discriminant (codegree of $(X,L,V)$) giving some bounds on it and classifying curves and surfaces of codegree 2 and 3. We exclude the possibility for t…
We revisit the abstract framework underlying the fibration method for producing rational points on the total space of fibrations over the projective line. By fine-tuning its dependence on external arithmetic conjectures, we render the method unconditional when the degree of the non-split locus is $\leq 2$, as well as in various instances where it is $3$. We are also able to obtain improved results in the regime that is conditionally accessible under Schinzel's hypothesis, by incorporating into it, for the first time, a technique due to Harari for controlling the Brauer--Manin obstruction in fa…
We prove that the Klein cubic threefold $F$ is the only smooth cubic threefold which has an automorphism of order 11. We compute the period lattice of the intermediate Jacobian of $F$ and study its Fano surface $S$. We compute also the set of fibrations of $S$ onto a curve of positive genus and the intersection between the fibres of these fibrations. These fibres generate an index 2 sub-group of the Néron-Severi group and we obtain a set of generators of this group. The Néron-Severi group of $S$ has rank $25=h^{1,1}$ and discriminant $11^{10}$.
Mirror of Orbifold Singularities in the Hitchin Fibration: the case $(\text{SL}_n,\text{PGL}_n)$
研究 Hitchin 纤维化中轨道奇点的镜像，证明 $\ ext{PGL}_n$ 模空间奇点对应 $\ ext{SL}_n$ 可约纤维，并验证 Fourier-Mukai 变换满足分数 Hecke 特征性质。
Let $f \colon (X,Δ) \to Y$ be a fibration such that $K_X + Δ$ is torsion along the fibres of $f$. Assume that $Y$ has dimension 2, or that $Y$ has dimension 3 and the fibres have dimension at most 3. Then the restriction of the moduli part to its augmented base locus is semiample.
In this paper we want to study the non-Kodaira fibres in a smooth equidimensional elliptic threefold. If the morphism to the Weierstrass model of the fibration is crepant, then we can locate the non-Kodaira fibres and give a description of their structure. In particular, they lie over the singular points of the (reduced) discriminant of the fibration and are contraction of a Kodaira fibre, whose type can be predicted using Tate's algorithm.
Equivariant Contact Darboux Quotients
本文建立了-1-移位接触导出阿廷栈的等变达布定理，并通过导出辛化与代数下降构造了ℓ-adic反常层，提取了导出枚举不变量，同时提出了接触版本的Joyce猜想并构造了范畴化勒让德2-范畴。
Special Kähler geometry and holomorphic Lagrangian fibrations
本文利用判别轨迹外特殊 Kähler 度量的微分几何，研究紧 hyperkähler 流形全纯 Lagrangian 纤维化的底空间，通过切丛拉回的平行分裂及既有结果重新推出其必为射影空间。
Optimal bound for singularities on Fano type fibrations of relative dimension one
本文针对相对维数为1的Fano型纤维化，在Shokurov猜想的基础上，给出了任意$0<ε≤1$时$(Z,B_Z+M_Z)$为$δ$-lc的最优$δ$值，完善了该领域的奇点最优界研究。
\special{html:<a href="hrefstring">} Let $Y$ be a del Pezzo variety of degree $d\leq 4$ and dimension $n\geq 3$, let $H$ be an ample class such that $-K_Y=(n-1)H$ and let $Z\subset Y$ be a $0$-dimensional subscheme of length $d$ such that the subsystem of elements of $|H|$ with base locus $Z$ gives a rational morphism $π_Z\colon Y\dashrightarrow{\mathbb P}^{n-1}$. Denote by $π\colon X\to {\mathbb P}^{n-1}$ the elliptic fibration obtained by resolving the indeterminacy locus of $π_Z$. Extending the results of [arXiv:1305.3340] we study the geometry of the variety $X$ and we prove that the Morde…
Fixed-locus relations and purely non-symplectic automorphisms of order 6 on K3 surfaces
本文研究K3曲面上6阶纯非辛自同构的固定轨迹关系，通过比较弦性欧拉特征与霍奇数推导关系，并利用局部条件将候选数从817减少到204，且给出20种固定轨迹的显式模型。
We study the collapsing behaviour of Ricci-flat Kahler metrics on a projective Calabi-Yau manifold which admits an abelian fibration, when the volume of the fibers approaches zero. We show that away from the critical locus of the fibration the metrics collapse with locally bounded curvature, and along the fibers the rescaled metrics become flat in the limit. The limit metric on the base minus the critical locus is locally isometric to an open dense subset of any Gromov-Hausdorff limit space of the Ricci-flat metrics. We then apply these results to study metric degenerations of families of pola…
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…
Let π: M \ra B be a Lagrangian torus fibration with singularities such that the fibers are of Maslov index zero, and unobstructed. The paper constructs a rigid analytic space M_0^\chk over the Novikov field which is a deformation of the semi-flat complex structure of the dual torus fibration over the smooth locus B_0 of π. Transition functions of M_0^\chk are obtained via A-\infty homomorphisms which capture the wall-crossing phenomenon of moduli spaces of holomorphic disks.
On strictly elliptic K3 surfaces and del Pezzo surfaces
本文利用 Alexeev-Nikulin 对应，分类了由光滑 del Pezzo 曲面二次曲线丛在特定 K3 曲面上诱导的椭圆纤维化，并研究了具有严格椭圆非辛对合的 K3 曲面上的线性系统。
Kodaira dimension of moduli of special $K3^{[2]}$-fourfolds of degree 2
该研究针对次数为2的$K3^{[2]}$-四形模空间的Noether-Lefschetz轨迹，计算了除有限个判别式外对应特殊除子的Kodaira维数，证明多数情形下该空间为一般型代数簇。
The Mumford--Tate group of a very general Lagrangian fiber
本文分类了本原辛簇上拉格朗日纤维化的非常一般纤维的Mumford--Tate群，推出判别式余维数一，并在超Kähler情形下证明Hodge-普遍性，证实了Li--Tosatti关于特殊Kähler度量正曲率的预期。
The SYZ Conjecture explains Mirror Symmetry between mirror Calabi-Yau 3-folds M,M' in terms of special Lagrangian fibrations f : M --> B and f' : M' --> B over the same base B, whose fibres are dual 3-tori, except for singular fibres. One of the main problems in proving the SYZ Conjecture (or even in finding the right statement of it) is that the singularities of special Lagrangian 3-folds and fibrations are poorly understood. This paper studies the singularities of special Lagrangian fibrations. Our main rigorous results are the construction of examples of special Lagrangian fibrations on ope…
We prove using symplectic field theory that if the suspension of a hyperbolic diffeomorphism of the two-torus Lagrangian embeds in a closed uniruled symplectic six-manifold, then its image contains the boundary of a symplectic disc with vanishing Maslov index. This prevents such a Lagrangian submanifold to be monotone, for instance the real locus of a smooth real Fano manifold. It also prevents any Sol manifold to be in the real locus of an orientable real Del Pezzo fibration over a curve, confirming an expectation of J. Kollár. Finally, it constraints Hamiltonian diffeomorphisms of uniruled s…
We interpret symplectic geometry as certain sheaf theory by constructing a sheaf of curved A_\infty algebras which in some sense plays the role of a "structure sheaf" for symplectic manifolds. An interesting feature of this "structure sheaf" is that the symplectic form itself is part of its curvature term. Using this interpretation homological mirror symmetry can be understood by well-known duality theories in mathematics: Koszul duality or Fourier- Mukai transform. In this paper we perform the above constructions over a small open subset inside the smooth locus of a Lagrangian torus fibration…
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