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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.
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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 on curves. There are many nice connections between stability conditions on $X$ and the geometry of the variety. We construct new stability conditions for surfaces containing a curve $C$ whose self-intersection is negative. We show that these stability conditions lie on a wall of the geometric chamber of ${rm Stab}(X)$, the stability manifold of $X$. We then construct the moduli space $M_{sigma}(mathcal{O}_X)$ of $sigma$-semistable objects of class $[mathcal{O}_X]$ in $K_0(X)$ after wall-crossing.
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Hara Charalambous
, Kostas Karagiannis
, Sotiris Karanikolopoulos andn Aristides Kontogeorgis
2021
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 ring $R=bigoplus H^0(F_n,Omega_{F_n}^{otimes m})$ when $p>3$, $p mid n$ and $n-1$ is not a power of $p$. In particular, we explicitly determine the classes $[H^0(F_n,Omega_{F_n}^{otimes m})]$ in the Grothendieck group $K_0(G,k)$ of finitely generated $k[G]$-modules, describe the respective equivariant Hilbert series $H_{R,G}(t)$ as a rational function, and use our results to write a program in Sage that computes $H_{R,G}(t)$ for an arbitrary Fermat curve.
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 $2lambda_h$ (an integer invariant associated to $h$). As one of the consequences of this, we prove that if $V_n$ is semistable in characteristic 0, then its reduction mod $p$ is strongly semistable, for $p$ in a Zariski dense set of primes. Moreover, for any given finitely many such semistable bundles $V_n$, there is a common Zariski dense set of such primes.
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 a strong HN (Harder-Narasimahan) slope of the canonical syzygy bundle on $mbox{Proj}~R$. Thus $c^I({bf m})$ is a rational number. This gives us a well defined notion, of the $F$-threshold $c^I({bf m})$ in characteristic $0$, in terms of a HN slope of the syzygy bundle on $mbox{Proj}~R$. This generalizes our earlier result (in [TrW]) where we have shown that if $I$ has homogeneous generators of the same degree, then the $F$-threshold $c^I({bf m})$ is expressed in terms of the minimal strong HN slope (in char $p$) and in terms of the minimal HN slope (in char $0$), respectively, of the canonical syzygy bundle on $mbox{Proj}~R$. Here we also prove that, for a given pair $(R, I)$ over a field of characteristic $0$, if $({bf m}_p, I_p)$ is a reduction mod $p$ of $({bf m}, I)$ then $c^{I_p}({bf m}_p) eq c^I_{infty}({bf m})$ implies $c^{I_p}({bf m}_p)$ has $p$ in the denominator, for almost all $p$.
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