arXiv subjects

Jaydeep Chipalkatti

Publications and source records attributed to Jaydeep Chipalkatti.

At least 19 recordsLinked to original sources

arXiv2023

On the Enumerative Geometry of Pascal's Hexagram

Given six points $A,B,C,D,E,F$ on a nonsingular conic in the complex projective plane, Pascal's theorem says that the three intersection points $AE \cap BF, BD \cap CE, AD \cap CF$ are collinear. The line containing them is called a pascal, and we get altogether $60$ such lines by permuting the points. In this paper, we consider the enumerative problem of finding the number of sextuples $(A, B, \dots, F)$ which correspond to three pre-specified pascals. We use computational techniques in commutative algebra to solve this problem in all cases. The results are tabulated using the so-called 'dual' notation for pascals, which is based upon the outer automorphism of $S_6$.

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arXiv2022

Degenerations of Pascal Lines

Let $\mathcal{K}$ denote a nonsingular conic in the complex projective plane. Pascal's theorem says that, given six distinct points $A,B,C,D,E,F$ on $\mathcal{K}$, the three intersection points $AE \cap BF, AD \cap CF, BD \cap CE$ are collinear. The line containing them is called the Pascal line of the sextuple. However, this construction may fail when some of the six points come together. In this paper, we find the indeterminacy locus where the Pascal line is not well-defined and then use blow-ups along polydiagonals to define it. We analyse the geometry of Pascals in these degenerate cases. Finally we offer some remarks about the indeterminacy of other geometric elements in Pascal's hexagrammum mysticum.

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arXiv2020

Absolute Projectivities in Pascal's Multimysticum

The Pascal Multimysticum is a system of points and lines constructed with a straight edge starting from six points on a conic. We show that the system contains 150 infinite ranges (and 150 infinite pencils) whose projective coordinates are absolutely fixed and independent of the conic and the hexagon that define the system.

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arXiv2019

Quadratic Involutions on Binary Forms

There is a classical geometric construction which uses a binary quadratic form to define an involution on the space of binary d-ics. We give a complete characterization of a general class of such involutions which are definable using compound transvectant formulae. We also study the associated varieties of forms which are preserved by such involutions. Along the way we prove a recoupling formula for transvectants, which is used to deduce a system of equations satisfied by the coefficients in these involutions.

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arXiv2017

On the dynamics of the Pappus-Steiner map

We extract a two-dimensional dynamical system from the theorems of Pappus and Steiner in classical projective geometry. We calculate an explicit formula for this system, and study its elementary geometric properties. Then we use Artin reciprocity to characterise all sufficiently large primes $p$ for which this system admits periodic points of orders $3$ and $4$ over the field ${\mathbb F}_p$; this leads to an unexpected Galois-theoretic conjecture for $n$-periodic points. We also give a short discussion of Leisenring's theorem, and show that it leads to the same dynamical system as the Pappus-Steiner theorem. The appendix contains a computer-aided analysis of this system over the field of real numbers.

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arXiv2017

On the letter frequencies and entropy of written Marathi

We carry out a comprehensive analysis of letter frequencies in contemporary written Marathi. We determine sets of letters which statistically predominate any large generic Marathi text, and use these sets to estimate the entropy of Marathi.

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arXiv2016

On the reconstruction problem for Pascal lines

Given a sextuple of distinct points $A, B, C, D, E, F$ on a conic, arranged into an array $\left[\begin{array}{ccc} A & B & C F & E & D \end{array}\right]$, Pascal's theorem says that the points $AE \cap BF, BD \cap CE, AD \cap CF$ are collinear. The line containing them is called the Pascal of the array, and one gets altogether sixty such lines by permuting the points. In this paper we prove that the initial sextuple can be explicitly reconstructed from four specifically chosen Pascals. The reconstruction formulae are encoded by some transvectant identities which are proved using the graphical calculus for binary forms.

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arXiv2016

On the geometry of the ricochet locus

This paper is a study of the so-called `ricochet configuration' (or $R$-configuration) which arises in the context of Pascal's theorem. We give a geometric proof of the fact that a specific pair of Pascal lines is coincident for a sextuple in $R$-configuration. We calculate the symmetry group of a generic $R$-configuration, as well as the degree of the subvariety ${\mathcal R} \subseteq {\mathbb P}^6$ of all such configurations. We also determine the $SL(2)$-equivariant defining equations for ${\mathcal R}$, and show that it is an ideal-theoretic complete intersection of two invariant hypersurfaces.

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arXiv2016

On the Poncelet triangle condition over finite fields

Let ${\mathbf P}^2$ denote the projective plane over a finite field ${\mathbb F}_q$. A pair of nonsingular conics $({\mathcal A}, {\mathcal B})$ in the plane is said to satisfy the Poncelet triangle condition if, considered as conics in ${\mathbf P}^2({\overline{\mathbb F}}_q)$, they intersect transverally and there exists a triangle inscribed in ${\mathcal A}$ and circumscribed around ${\mathcal B}$. It is shown in this article that a randomly chosen pair of conics satisfies the triangle condition with asymptotic probability $1/q$. We also make a conjecture based upon computer experimentation which predicts this probability for tetragons, pentagons and so on up to enneagons.

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arXiv2015

When are the Cayley-Salmon lines conjugate?

Given six points on a conic, Pascal's theorem gives rise to a well-known configuration called the \emph{hexagrammum mysticum}. It consists of, amongst other things, twenty Steiner points and twenty Cayley-Salmon lines. It is a classical theorem due to von Staudt that the Steiner points fall into ten conjugate pairs with reference to the conic; but this is not true of the C-S lines for a general choice of six points. It is shown in this paper that the C-S lines are pairwise conjugate precisely when the original sextuple is~\emph{tri-involutive}. The variety of tri-involutive sextuples turns out to be arithmetically Cohen-Macaulay of codimension two. We determine its $SL_2$-equivariant minimal resolution.

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arXiv2014

On the coincidence of Pascal lines

Let ${\mathcal K}$ denote a smooth conic in the complex projective plane. Pascal's theorem says that, given six points $A,B,C,D,E,F$ on ${\mathcal K}$, the three intersection points $AE \cap BF, AD \cap CF, BD \cap CE$ are collinear. This defines the Pascal line of the array $\left[ \begin{array}{ccc} A & B & C \\ F & E & D \end{array} \right]$, and one gets sixty such lines in general by permuting the points. In this paper we consider the variety $Ψ$ of sextuples $\{A, \dots, F\}$, for which some of these Pascal lines coincide. We show that $Ψ$ has two irreducible components: a five-dimensional component of sextuples in involution, and a four-dimensional component of the so-called `ricochet configurations'. This gives a complete synthetic characterisation of points in $Ψ$. The proof relies upon Gröbner basis techniques to solve multivariate polynomial equations.

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arXiv2012

On Hilbert covariants

Let F denote a binary form of order d over the complex numbers. If r is a divisor of d, then the Hilbert covariant H_{r,d}(F) vanishes exactly when F is the perfect power of an order r form. In geometric terms, the coefficients of H give defining equations for the image variety X of an embedding P^r->P^d. In this paper we describe a new construction of the Hilbert covariant; and simultaneously situate it into a wider class of covariants called the Göttingen covariants, all of which vanish on X. We prove that the ideal generated by the coefficients of H defines X as a scheme. Finally, we exhibit a generalisation of the Göttingen covariants to n-ary forms using the classical Clebsch transfer principle.

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arXiv2009

On the saturation sequence of the rational normal curve

Let $C \subseteq ¶^d$ denote the rational normal curve of order $d$. Its homogeneous defining ideal $I_C \subseteq \QQ[a_0,...,a_d]$ admits an $SL_2$-stable filtration $J_2 \subseteq J_4 \subseteq ... \subseteq I_C$ by sub-ideals such that the saturation of each $J_{2q}$ equals $I_C$. Hence, one can associate to $d$ a sequence of integers $(α_1,α_2,...)$ which encodes the degrees in which the successive inclusions in this filtration become trivial. In this paper we establish several lower and upper bounds on the $α_q$, using \emph{inter alia} the methods of classical invariant theory.

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arXiv2009

On the ideals of general binary orbits

Let $E$ denote a general complex binary form of order $d$ (seen as a point in $¶^d$), and let $Ω_E \subseteq ¶^d$ denote the closure of its $SL_2$-orbit. In this note, we calculate the equivariant minimal generators of its defining ideal $I_E \subseteq \complex[a_0,...,a_d]$ for $4 \leqslant d \leqslant 10$. In order to effect the calculation, we introduce a notion called the `graded threshold character' of $d$. One unexpected feature of the problem is the (rare) occurrence of the so-called `invisible' generators in the ideal, and the resulting dichotomy on the set of integers $d \geqslant 4$.

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arXiv2008

On the Linear Combinants of a Binary Pencil

Let A,B denote binary forms of order d, and let C_{2r-1} = (A,B)_{2r-1} be the sequence of their linear combinants for r between 1 and (d+1)/2. It is known that C_1 and C_3 together determine the pencil generated by A and B, and hence indirectly the higher C_{2r-1}. In this paper we exhibit explicit formulae for all r>2, which allow us to recover C_{2r-1} from the knowledge of C_1 and C_3. The calculations make use of the symbolic method of classical invariant theory, as well as the quantum theory of angular momentum. Our theorem pertains to the second exterior power representation of S_d, for the group SL_2. We give an example for the group SL_3 to show that such a result may hold for other categories of representations.

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arXiv2008

The Higher Transvectants are Redundant

Let A, B denote generic binary forms, and let u_r = (A,B)_r denote their r-th transvectant in the sense of classical invariant theory. In this paper we classify all the quadratic syzygies between the u_r. As a consequence, we show that each of the higher transvectants u_r, r>1, is redundant in the sense that it can be completely recovered from u_0 and u_1. This result can be geometrically interpreted in terms of the incomplete Segre imbedding. The calculations rely upon the Cauchy exact sequence of SL_2-representations, and the notion of a 9-j symbol from the quantum theory of angular momentum. We give explicit computational examples for SL_3, g_2 and S_5 to show that this result has possible analogues for other categories of representations.

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arXiv2006

On Hermite's invariant for binary quintics

The Hermite invariant H is the defining equation for the hypersurface of binary quintics in involution. This paper analyses the geometry and invariant theory of H. We determine the singular locus of this hypersurface and show that it is a complete intersection of a linear covariant of quintics. The projective dual of this hypersurface can be identified with itself via an involution. It is shown that the Jacobian ideal of H is perfect of height two, and we describe its SL_2-equivariant minimal resolution. The last section develops a general formalism for evectants of covariants of binary forms, which is then used to calculate the evectant of H.

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arXiv2006

On the Jacobian ideal of the binary discriminant

Let $Δ$ denote the discriminant of a generic binary $d$-ic. We show that for $d \ge 3$, the Jacobian ideal of $Δ$ is perfect of height 2. Moreover, we describe its SL_2-equivariant minimal resolution, and the associated invariant differential equations satisfied by $Δ$. A similar result is proved for the resultant of two forms of orders $d,e$, whenever $d \ge e-1$. We also explain the role of the Morley form in the determinantal formula for the resultant; this relies upon a calculation which is done in the appendix by A. Abdesselam.

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