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Register a new userTschirnhaus-Weierstrass curves
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Added by
David Sevilla
Publication date
2011
fields
Informatics Engineering
and research's language is
English
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We define the concept of Tschirnhaus-Weierstrass curve, named after the Weierstrass form of an elliptic curve and Tschirnhaus transformations. Every pointed curve has a Tschirnhaus-Weierstrass form, and this representation is unique up to a scaling of variables. This is useful for computing isomorphisms between curves.
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We prove equidistribution of Weierstrass points on Berkovich curves. Let $X$ be a smooth proper curve of positive genus over a complete algebraically closed non-Archimedean field $K$ of equal characteristic zero with a non-trivial valuation. Let $L$ be a line bundle of positive degree on $X$. The Weierstrass points of powers of $L$ are equidistributed according to the Zhang-Arakelov measure on the analytification $X^{an}$. This provides a non-Archimedean analogue of a theorem of Mumford and Neeman. Along the way we provide a description of the reduction of Weierstrass points, answering a question of Eisenbud and Harris.
Let RD(n) denote the minimum d for which there exists a formula for the roots of the general degree n polynomial using only algebraic functions of d or fewer variables. In 1927, Hilbert sketched how the 27 lines on a cubic surface could be used to construct a 4-variable formula for the general degree 9 polynomial (implying $RD(9)le 4$). In this paper, we turn Hilberts sketch into a general method. We show this method produces best-to-date upper bounds on RD(n) for all n, improving earlier results of Hamilton, Sylvester, Segre and Brauer.
Let $C$ be a hyperelliptic curve of genus $g$ over the fraction field $K$ of a discrete valuation ring $R$. Assume that the residue field $k$ of $R$ is perfect and that $mathop{textrm{char}} k eq 2$. Assume that the Weierstrass points of $C$ are $K$-rational. Let $S = mathop{textrm{Spec}} R$. Let $mathcal{X}$ be the minimal proper regular model of $C$ over $S$. Let $mathop{textrm{Art}} (mathcal{X}/S)$ denote the Artin conductor of the $S$-scheme $mathcal{X}$ and let $ u (Delta)$ denote the minimal discriminant of $C$. We prove that $-mathop{textrm{Art}} (mathcal{X}/S) leq u (Delta)$. As a corollary, we obtain that the number of components of the special fiber of $mathcal{X}$ is bounded above by $ u(Delta)+1$.
We analyze Weierstrass cycles and tautological rings in moduli space of smooth algebraic curves and in moduli spaces of integral algebraic curves with embedded disks with special attention to moduli spaces of curves having genus $leq 6$. In particular, we show that our general formula gives a good estimate for the dimension of Weierstrass cycles for lower genera.
In this work we study the generalized Weierstrass semigroup $widehat{H} (mathbf{P}_m)$ at an $m$-tuple $mathbf{P}_m = (P_{1}, ldots , P_{m})$ of rational points on certain curves admitting a plane model of the form $f(y) = g(x)$ over $mathbb{F}_{q}$, where ${f(T),g(T)in mathbb{F}_q[T]}$. In particular, we compute the generating set $widehat{Gamma}(mathbf{P}_m)$ of $widehat{H} (mathbf{P}_m)$ and, as a consequence, we explicit a basis for Riemann-Roch spaces of divisors with support in ${P_{1}, ldots , P_{m}}$ on these curves, generalizing results of Maharaj, Matthews, and Pirsic.
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