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Let $X$ be the blowup of a weighted projective plane at a general point. We study the problem of finite generation of the Cox ring of $X$. Generalizing examples of Srinivasan and Kurano-Nishida, we consider examples of $X$ that contain a negative curve of the class $H-mE$, where $H$ is the class of a divisor pulled back from the weighted projective plane and $E$ is the class of the exceptional curve. For any $m>0$ we construct examples where the Cox ring is finitely generated and examples where it is not.
We develop a method to compute limits of dual plane curves in Zeuthen families of any kind. More precisely, we compute the limit 0-cycle of the ramification scheme of a general linear system on the generic fiber, only assumed geometrically reduced, of a Zeuthen family of any kind.
Let $X$ be a smooth complex projective variety. In 2002, Bridgeland defined a notion of stability for the objects in $D^b(X)$, the bounded derived category of coherent sheaves on $X$, which generalized the notion of slope stability for vector bundles
We consider a Fermat curve $F_n:x^n+y^n+z^n=1$ over an algebraically closed field $k$ of characteristic $pgeq0$ and study the action of the automorphism group $G=left(mathbb{Z}/nmathbb{Z}timesmathbb{Z}/nmathbb{Z}right)rtimes S_3$ on the canonical rin
Here we consider the set of bundles ${V_n}_{ngeq 1}$ associated to the plane trinomial curves $k[x,y,z]/(h)$. We prove that the Frobenius semistability behaviour of the reduction mod $p$ of $V_n$ is a function of the congruence class of $p$ modulo $2
We show that if $R$ is a two dimensional standard graded ring (with the graded maximal ideal ${bf m}$) of characteristic $p>0$ and $Isubset R$ is a graded ideal with $ell(R/I) <infty$ then the $F$-threshold $c^I({bf m})$ can be expressed in terms of