On real determinantal quartics

Determinantal quartics and the computation of the Picard group

We test the methods for computing the Picard group of a $K3$ surface in a situation of high rank. The examples chosen are resolutions of quartics in $\bP^3$ having 14 singularities of type $A_1$. Our computations show that the method of R. van Luijk works well when sufficiently large primes are used.

On algorithms to obtain linear determinantal representations of smooth plane curves of higher degree

We give two algorithms to compute linear determinantal representations of smooth plane curves of any degree over any field. As particular examples, we explicitly give representatives of all equivalence classes of linear determinantal representations of two special quartics over the field $\mathbb{Q}$ of rational numbers, the Klein quartic and the Fermat quartic. This paper is a summary of third author's talk at the JSIAM JANT workshop on algorithmic number theory in March 2018. Details will appear elsewhere.

Quarto-quartic birational maps of $\mathbb{P}_3(\mathbb{C})$

We construct a determinantal family of quarto-quartic transformations of a complex projective space of dimension $3$ from trigonal curves of degree $8$ and genus $5$. Moreover we show that the variety of $(4,4)$-birational maps of $\mathbb{P}_3(\mathbb{C})$ has at least four irreducible components and describe three of them.

Livsic-type Determinantal Representations and Hyperbolicity

Hyperbolic homogeneous polynomials with real coefficients, i.e., hyperbolic real projective hypersurfaces, and their determinantal representations, play a key role in the emerging field of convex algebraic geometry. In this paper we consider a natural notion of hyperbolicity for a real subvariety $X \subset \mathbb{P}^d$ of an arbitrary codimension $\ell$ with respect to a real $\ell - 1$-dimensional linear subspace $V \subset \mathbb{P}^d$ and study its basic properties. We also consider a special kind of determinantal representations that we call Livsic-type and a nice subclass of these that…

Geometric classification of real ternary octahedral quartics

Ternary real-valued quartics in $\mathbb{R}^3$ being invariant under octahedral symmetry are considered. The geometric classification of these surfaces is given. A new type of surfaces emerge from this classification.

四次光滑曲面上的圆锥曲线

Conics on smooth quartic surfaces

研究了光滑四次曲面上圆锥曲线的最大数量，证明其最大值为800，并对具有大量圆锥曲线的四次曲线进行了分类。

Real representatives of equisingular strata of simple quartic surfaces

We develop an algorithm detecting real representatives in equisingular strata of projective models of $K3$-surfaces. We apply this algorithm to spatial quartics and find two new examples of real strata without real representatives. As a by-product, we also give a new proof for the only previously known example of plane sextics.

Singular points of real quartic curves via computer algebra

There are thirteen types of singular points for irreducible real quartic curves and seventeen types of singular points for reducible real quartic curves. This classification is originally due to D.A. Gudkov. There are nine types of singular points for irreducible complex quartic curves and ten types of singular points for reducible complex quartic curves. We derive the complete classification with proof by using the computer algebra system Maple. We clarify that the classification is based on computing just enough of the Puiseux expansion to separate the branches. Thus, the proof consists of a…

Lines on quartic surfaces

We show that the maximal number of (real) lines in a (real) nonsingular spatial quartic surface is 64 (respectively, 56). We also give a complete projective classification of all quartics containing more than 52 lines: all such quartics are projectively rigid. Any value not exceeding 52 can appear as the number of lines of an appropriate quartic.

平面四次曲线实切线的避开轨迹与热带化

Avoidance loci and tropicalizations of real bitangents to plane quartics

本文比较了光滑平面四次曲线实切线的两种4条一组的划分方式，并利用Tarski原理、实平面四次曲线拓扑及热带几何证明二者在均有定义时完全一致。

On quartic forms associated with cubic transformations of the real plane

A polynomial transformation of the real plane $\Bbb R^2$ is a mapping $\Bbb R^2\to\Bbb R^2$ given by two polynomials of two variables. Such a transformation is called cubic if the degrees of its polynomials are not greater than three. It turns out that cubic transformations are associated with some binary and quaternary quartic forms. In the present paper these forms are defined and studied.

Moduli of real pointed quartic curves

We describe a natural open stratum in the moduli space of smooth real pointed quartic curves in the projective plane. This stratum consists of real isomorphism classes of pairs $(C, p)$ with $p$ a real point on the curve $C$ such that the tangent line at $p$ intersects the curve in two distinct points besides $p$. We will prove that this stratum consists of $20$ connected components. Each of these components has a real toric structure defined by an involution in the Weyl group of type $E_7$.

界定平面四次曲线实双切线的带符号计数

Bounding the signed count of real bitangents to plane quartics

针对实光滑平面四次曲线实双切线的带符号计数，本文运用实代数几何、平面几何与凸集性质，证明了Larson和Vogt提出的其上界为8的猜想。

关于平面实四次曲线与三条直线的排列

On arrangements of plane real quartics with respect to three lines

研究光滑实代数或实伪全纯四次曲线与三条直线的相互排列分类，在特定条件下完成分类。有一种排列可伪全纯但非代数实现，通过组合拼接构建，是此类产生代数不可实现PL曲线的首个例子。

On minimality of determinantal varieties

We prove that semialgebraic sets of rectangular matrices of a fixed rank, of skew-symmetric matrices of a fixed rank and of real symmetric matrices whose eigenvalues have prescribed multiplicities are minimal submanifolds of the space of real matrices of a given size.

对称行列式簇的实与正热带化

Real and Positive Tropicalizations of Symmetric Determinantal Varieties

本文研究了Puiseux级数上低秩对称矩阵簇的实与正热带化，证明了实与复热带化在秩二和余秩一情形下一致，并揭示了Speyer和Williams提出的两种正热带化概念在相同条件下的异同。

Rationality of real conic bundles with quartic discriminant curve

We study real double covers of $\mathbb P^1\times\mathbb P^2$ branched over a $(2,2)$-divisor, which have the structure of a conic bundle threefold with smooth quartic discriminant curve via the second projection. In each isotopy class of smooth plane quartics, we construct examples where the total space of the conic bundle is rational. For five of the six isotopy classes we construct $\mathbb C$-rational examples that have obstructions to rationality over $\mathbb R$, and for the sixth class, we show that the models we consider are all rational. Moreover, for three of the five classes with ir…

Computing Hermitian determinantal representations of hyperbolic curves

Every real hyperbolic form in three variables can be realized as the determinant of a linear net of Hermitian matrices containing a positive definite matrix. Such representations are an algebraic certificate for the hyperbolicity of the polynomial and their existence has been proved in several different ways. However, the resulting algorithms for computing determinantal representations are computationally intensive. In this note, we present an algorithm that reduces a large part of the problem to linear algebra and discuss its numerical implementation.

关于平面实曲线与M-四次曲线的相互安排

On mutual arrangements of a plane real curve relative to an $M$-quartic with an oval-snake

本文研究了平面实曲线与M-四次曲线的相互安排，证明了在特定条件下，实曲线与四次曲线的拓扑等价性，并探讨了伪全纯曲线的类似结论。

A New Proof of Hilbert's Theorem on Ternary Quartics

David Hilbert proved that a non-negative real quartic form f(x,y,z) is the sum of three squares of quadratic forms. We give a new proof which shows that if the complex plane curve Q defined by f is smooth, then f has exactly 8 such representations, up to equivalence. They correspond to those real 2-torsion points of the Jacobian of Q which are not represented by a conjugation-invariant divisor on Q.

实确定性变种的切锥：多种表达式及证明

The tangent cone to the real determinantal variety: various expressions and a proof

本文指出前人关于实确定性变种切锥的证明不完整，并提供新证明，同时回顾了切锥的等价描述，证明该变种的切锥与代数切锥一致，这在一般实代数变种中不成立。

Determinantal representations of hyperbolic plane curves: An elementary approach

If a real symmetric matrix of linear forms is positive definite at some point, then its determinant is a hyperbolic hypersurface. In 2007, Helton and Vinnikov proved a converse in three variables, namely that every hyperbolic plane curve has a definite real symmetric determinantal representation. The goal of this paper is to give a more concrete proof of a slightly weaker statement. Here we show that every hyperbolic plane curve has a definite determinantal representation with Hermitian matrices. We do this by relating the definiteness of a matrix to the real topology of its minors and extendi…

Real Quartic Surfaces in RP_3 containing 16 Skew Lines

In [BN] the authors construct a special complex of degree 20 over M, which for an open three dimensional set parametrizes smooth complex surfaces of degree four invariant which are Heisenberg invariant and each member of the family contains 32 lines but only 16 skew lines. The coordinates of the lines however need not be real. For a point l in a Zariski open set of M an algorithm is presented which evaluates the real coefficients of l in terms of the K-coordinates of l. The author uses a code in Maple which allows him to construct very explicite examples of real smooth Heisenberg invariant Kum…

Determinantal representations of hyperbolic curves via polynomial homotopy continuation

A smooth curve in the real projective plane is hyperbolic if its ovals are maximally nested. By the Helton-Vinnikov Theorem, any such curve admits a definite symmetric determinantal representation. We use polynomial homotopy continuation to compute such representations numerically. Our method works by lifting paths from the space of hyperbolic polynomials to a branched cover in the space of pairs of symmetric matrices.

实理性四次德尔皮佐表面的自同构群

Automorphism groups of real rational quartic del Pezzo surfaces

本文通过研究实理性四次德尔皮佐表面作为某些光滑实二次曲面的吹胀，描述了所有可能的自同构群，并利用其对复结构的Gal(C/R)作用，给出了实自同构群的几何描述。

Generic pointed quartic curves in $\mathbb{R}\mathbb{P}^{2}$ and uninodal dessins

In this article we obtain a rigid isotopy classification of generic pointed quartic curves $(A,p)$ in $\mathbb{R}\mathbb{P}^{2}$ by studying the combinatorial properties of dessins. The dessins are real versions, proposed by S. Orevkov, of Grothendieck's dessins d'enfants. This classification contains 20 classes determined by the number of ovals of $A$, the parity of the oval containing the marked point $p$, the number of ovals that the tangent line $T_p A$ intersects, the nature of connected components of $A\setminus T_p A$ adjacent to $p$, and in the maximal case, on the convexity of the pos…

Root Configurations of Real Univariate Cubics and Quartics

For the general monic cubic and quartic with real coefficients, polynomial conditions on the coefficients are derived as directly and as simply as possible from the Sturm sequence that will determine the real and complex root multiplicities together with the order of the real roots with respect to multiplicity.

A tropical count of real bitangents to plane quartic curves

A smooth tropical quartic curve has seven tropical bitangent classes. Their shapes can vary within the same combinatorial type of curve. We study deformations of these shapes and we show that the conditions determined by Cueto and Markwig for lifting them to real bitangent lines are independent of the deformations. From this we deduce a tropical proof of Plücker and Zeuthen's count of the number of real bitangents to smooth plane quartic curves.

实热带四次曲线及其双切线：用补片构造计数

Real Tropical Quartics and their Bitangents: Counting with Patchworking

本文利用补片构造技术解决了特定热带四次曲线双切线实数提升行为的预测问题，并分析了实热带四次曲线的组合类型，附有polymake扩展及polyDB数据。

组合学与热带四次曲线双切线的实提升

Combinatorics and real lifts of bitangents to tropical quartic curves

本文研究热带平面四次曲线双切线的七个等价类，给出其41种类型的完整组合分类，并建立显式符号规则以构造实提升，证实每类恰有零个或四个全实提升。

Computing tropical bitangents to smooth quartic curves in polymake

In this article we introduce the recently developed polymake extension TropicalQuarticCurves and its associated database entry in polyDB dealing with smooth tropical quartic curves. We report on algorithms implemented to analyze tropical bitangents and their lifting conditions over real closed valued fields. The new functions and data were used by the authors to provide a tropical proof of Plücker and Zeuthen's count of real bitangents to smooth quartic curves.

Rational Quartic Spectrahedra

Rational quartic spectrahedra in 3-space are semialgebraic convex subsets in $\mathbb{R}^3$ of semidefinite, real symmetric $(4 \times 4)$-matrices, whose boundary admits a rational parameterization. The Zariski closure in $\mathbb{C}\mathbb{P}^3$ of the boundary of a rational spectrahedron is a rational complex symmetroid. We give necessary conditions on the configurations of singularities of the corresponding real symmetroids in $\mathbb{R}\mathbb{P}^3$ of rational quartic spectrahedra. We provide an almost exhaustive list of examples realizing the configurations, and conjecture that the mis…

Special geometry of quartic curves

We classify maximal quartic generalised projective special real curves up to equivalence. A maximal quartic generalised projective special real curve consists of connected components of the intersection of the hyperbolic points of a quartic homogeneous real polynomial $h:\mathbb{R}^2\to\mathbb{R}$ and its level set $\{h=1\}$. Two such curves are called equivalent if they are related by a linear coordinate transformation. As an application of our results we prove that quartic generalised projective special real manifolds that are homogeneous spaces have non-regular boundary behaviour, meaning t…

Conics Touching a Quartic Surface with 13 Nodes

Motivated by questions occuring in the construction of certain twistor spaces the parameter space of conics tangent to a given quartic is investigated. For a given real quartic surface in complex $\PP ^3$ that has exactly 13 ordinary nodes as singularities this parameter space is studied. In particular, its irreducible components are described.

Quartic Spectrahedra

Quartic spectrahedra in 3-space form a semialgebraic set of dimension 24. This set is stratified by the location of its ten nodes. There are twenty maximal strata, identified recently by Degtyarev and Itenberg, via the global Torelli Theorem for real K3 surfaces. We here give a new proof that is self-contained and algorithmic. This involves extending Cayley's characterization of quartic symmetroids, by the property that the branch locus of the projection from a node consists of two cubic curves. This paper represents a first step towards the classification of all spectrahedra of a given degree…