The minimal degree of plane models of double covers of smooth curves

If $X$ is a smooth curve such that the minimal degree of its plane models is not too small compared with its genus, then $X$ has been known to be a double cover of another smooth curve $Y$ under some mild condition on the genera. However there are no results yet for the minimal degrees of plane models of double covers except some special cases. In this paper, we give upper and lower bounds for the minimal degree of plane models of the double cover $X$ in terms of the gonality of the base curve $Y$ and the genera of $X$ and $Y$. In particular, the upper bound equals to the lower bound in case $Y$ is hyperelliptic. We give an example of a double cover which has plane models of degree equal to the lower bound.Comment: 13 pages; Sharpened the main result (Theorem 3.8); Corrected some errors (Theorem 4.1); Final version to appear in JPA

The support of top graded local cohomology modules

Let $R_0$ be any domain, let $R=R_0[U_1, ..., U_s]/I$, where $U_1, ..., U_s$ are indeterminates of some positive degrees, and $I\subset R_0[U_1, ..., U_s]$ is a homogeneous ideal. The main theorem in this paper is states that all the associated primes of $H:=H^s_{R_+}(R)$ contain a certain non-zero ideal $c(I)$ of $R_0$ called the ``content'' of $I$. It follows that the support of $H$ is simply V(\content(I)R + R_+) (Corollary 1.8) and, in particular, $H$ vanishes if and only if $c(I)$ is the unit ideal. These results raise the question of whether local cohomology modules have finitely many minimal associated primes-- this paper provides further evidence in favour of such a result. Finally, we give a very short proof of a weak version of the monomial conjecture based on these results

Weighted monotonicity theorems and applications to minimal surfaces in hyperbolic space

We show that there is a weighted version of monotonicity theorem corresponding to each function on a Riemannian manifold whose Hessian is a multiple of the metric tensor. Such function appears in the Euclidean space, the hyperbolic space $\mathbb{H}^n$ and the round sphere $\mathbb{S}^n$ as the distance function, the Minkowskian coordinates of $\mathbb{R}^{n,1}$ and the Euclidean coordinates of $\mathbb{R}^{n+1}$. In $\mathbb{H}^n$, we show that the time-weighted monotonicity theorem implies the unweighted version in \cite{Anderson82}. Applications include upper bounds for Graham--Witten renormalised area of minimal surfaces in term of the length of boundary curve and a complete computation of Alexakis--Mazzeo degrees defined in \cite{Alexakis.Mazzeo10}. An argument on area-minimising cones suggests the existence of a minimal surface in $\mathbb{H}^4$ bounded by the Hopf link $\{zw=\epsilon > 0, |z|^2 +|w|^2 = 1\}$ other than the pair of disks. We give an explicit construction of a minimal annulus in $\mathbb{H}^4$ with this property and obtain by the same method its sister in $\mathbb{S}^4$. A weighted monotonicity theorem is also proved in Riemannian manifolds whose sectional curvature is bounded from above.Comment: 22 pages. Cor 30, Rem 32, Prop 46 added. Overlap with Choe--Gulliver acknowledged. Minor upgrade of Comparison Lemm

Some minimisation algorithms in arithmetic invariant theory

We extend the work of Cremona, Fisher and Stoll on minimising genus one curves of degrees 2,3,4,5, to some of the other representations associated to genus one curves, as studied by Bhargava and Ho. Specifically we describe algorithms for minimising bidegree (2,2)-forms, 3 x 3 x 3 cubes and 2 x 2 x 2 x 2 hypercubes. We also prove a theorem relating the minimal discriminant to that of the Jacobian elliptic curve

Combinatorial Properties of Polyiamonds

Polyiamonds are plane geometric figures constructed by pasting together equilateral triangles edge-to-edge. It is shown that a diophantine equation involving vertices of degrees 2, 3, 5 and 6 holds for all polyiamonds; then an Eberhard-type theorem is proved, showing that any four-tuple of non-negative integers that satisfies the diophantine equation can be realized geometrically by a polyiamond. Further combinatorial and graph-theoretic aspects of polyiamonds are discussed, including a characterization of those polyiamonds that are three-connected and so three-polytopal, a result on Hamiltonicity, and constructions that use minimal numbers of triangles in realizing four-vectors