Expository Papers

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Here is a collection of notes and expository papers I've written over the last few years. Please note that the majority of these papers began their lives as personal notes, so let me know via email if the writing is not so clear!

Geometry

A neat way to compute the lines contained in a projective hypersurface.
A note on (12₃)-configurations from Grünbaum's book, including a word on deficiency complices.
Exterior Products and Plücker Coordinates.
An outline of computing the canonical divisor of ℙ² using an explicit by-definition construction.
Some of my notes from first learning about Chern classes and Schubert cycles.
A short demo on why the group law of a nodal cubic curve is multiplicative (and why the nodal cubic is rational).
A dive into Weierstrass points on an algebraic curve.
How points and tangent lines can determine conics in ℙ². This explains some of the gadgets in my gallery.
The 27 lines on a cubic surface are the complete intersection of the cubic with a nonic surface, proven using divisors.
Some notes on the Weil Pairing.
Some notes on the Weyl Group and how it relates to Entropy.
My notes on the space of complete conics, as I was reading Eisenbud and Harris' 3264 and all that Intersection Theory.
A brief intro to blowing up a projective plane (assuming you are familiar with blowing up an affine plane), and why ℙ² blown up at two points is a quadric surface in ℙ³.
A geometric look into the group Pic⁰ of a quartic curve in a rather stream-of-conscious style.
A look at the rational equivalence classes of the tangent and secant varieties of the twisted cubic, viewed as subvarieties of Gr(2,4).
How the orthocanonical subgroup of the Picard group of an elliptic fibration can show us the subgroup generated by the basepoints on the fibers.
A recipe for constructing an octic curve with seven triple points, with a link to a (very slow) Desmos demonstration.
A description of the Double Plane Cover induced by the Chilean configuration of points, and an aside about special fibers on elliptic surfaces.
A very brief word on morphisms defined by linear systems, with a view toward hyperelliptic curves.
Going through some of Dolgachev's examples of configurations and designs in projective geometry.
Computing the automorphism group of the projective plane blown up at four general points.
An exercise on why tangent lines are unique for nonsingular points, and why the Hessian coincides with flex points.

Algebra

Combinatorics

Some of my design theory notes.
An explicit construction of a maximal partial spread in ℙ³_𝔽₁₁.
A description of spreads in ℙ⁵, with a postscript describing Gorla's general construction of spreads using companion matrices.
A description of the Chow Rings of the Grassmannians Gr(3,5) and Gr(3,6), including a count of the number of Schubert cycles over 𝔽₂.

Probability

Calculating the probabilities of AB and AA appearing in a string of n random Latin letters.

Differential Equations

Rock-Paper-Scissors style systems of differential equations on 3, 5, and 7 functions.

Logic

My notes on t-norms and conorms and fuzzy logic.

Math Puzzles

A variation of the 12-coins puzzle.