Spin Curves and Scorza Quartics

Scorza Quartics of Trigonal Spin Curves and Their Varieties of Power Sums

Our fundamental result is the construction of new subvarieties in the varieties of power sums for the Scorza quartic of any general pairs of trigonal curves and non-effective theta characteristics. This is a generalization of Mukai’s description of smooth prime Fano threefolds of genus twelve as the varieties of power sums for plane quartics. Among other applications, we give an affirmative ans...

On Blow-ups of the Quintic Del Pezzo 3-fold and Varieties of Power Sums of Quartic Hypersurfaces

We construct new subvarieties in the varieties of power sums for certain quartic hypersurfaces. This provides a generalization of Mukai’s description of smooth prime Fano threefolds of genus twelve as the varieties of power sums for plane quartics. In fact in [TZ08] we show that these quartics are exactly the Scorza quartics associated to general pairs of trigonal curves and ineffective theta c...

Divison Polynomials for Alternate Models of Elliptic Curves

In this paper we find division polynomials for Huff curves, Jacobi quartics, and Jacobi intersections. These curves are alternate models for elliptic curves to the more common Weierstrass curve. Division polynomials for Weierstrass curves are well known, and the division polynomials we find are analogues for these alternate models. Using the division polynomials, we show recursive formulas for ...

Explicit equations in the plane for elliptic curves given as space quartics

On the computation of Mordell-Weil and 2-Selmer Groups of Elliptic Curves

The first provides an alternative to computing the height pairing matrix of the given set of points and showing that its determinant is non-zero. While that is easily done, for curves of large rank it requires some delicate consideration of precision in order to be sure of the result. The method here, by contrast, involves only “discrete” computations: finding roots of cubics and evaluating qua...