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Register a new userGeneralized Weierstrass semigroups and Riemann-Roch spaces for certain curves with separated variables
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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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We investigate the structure of the generalized Weierstrass semigroups at several points on a curve defined over a finite field. We present a description of these semigroups that enables us to deduce properties concerned with the arithmetical structure of divisors supported on the specified points and their corresponding Riemann-Roch spaces. This characterization allows us to show that the Poincare series associated with generalized Weierstrass semigroups carry essential information to describe entirely their respective semigroups.
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.
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.
We produce a Grothendieck transformation from bivariant operational $K$-theory to Chow, with a Riemann-Roch formula that generalizes classical Grothendieck-Verdier-Riemann-Roch. We also produce Grothendieck transformations and Riemann-Roch formulas that generalize the classical Adams-Riemann-Roch and equivariant localization theorems. As applications, we exhibit a projective toric variety $X$ whose equivariant $K$-theory of vector bundles does not surject onto its ordinary $K$-theory, and describe the operational $K$-theory of spherical varieties in terms of fixed-point data. In an appendix, Vezzosi studies operational $K$-theory of derived schemes and constructs a Grothendieck transformation from bivariant algebraic $K$-theory of relatively perfect complexes to bivariant operational $K$-theory.
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$.
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