Recent zbMATH articles in MSC 14H50https://www.zbmath.org/atom/cc/14H502021-04-16T16:22:00+00:00WerkzeugBounds for the number of points on curves over finite fields.https://www.zbmath.org/1456.111132021-04-16T16:22:00+00:00"Arakelian, Nazar"https://www.zbmath.org/authors/?q=ai:arakelian.nazar"Borges, Herivelto"https://www.zbmath.org/authors/?q=ai:borges.heriveltoSummary: Let \(\mathcal{X}\) be a projective irreducible nonsingular algebraic curve defined over a finite field \(\mathbb{F}_q\). This paper presents a variation of the Stöhr-Voloch theory and sets new bounds to the number of \(\mathbb{F}_{q^r}\)-rational points on \(\mathcal{X}\). In certain cases, where comparison is possible, the results are shown to improve other bounds such as Weil's, Stöhr-Voloch's and Ihara's.Hypersurfaces with linear type singular loci.https://www.zbmath.org/1456.130082021-04-16T16:22:00+00:00"Farrahy, Amir Behzad"https://www.zbmath.org/authors/?q=ai:farrahy.amir-behzad"Nasrollah Nejad, Abbas"https://www.zbmath.org/authors/?q=ai:nasrollah-nejad.abbasThe authors consider an affine hypersurface \(X\subset\mathbb A^n\) defined by a polynomial \(f\) in the ring \(R=k[x_1,\dots,x_n]\) over an algebraically closed field \(k\) of characteristic \(0\). The singular locus of \(X\) is defined by the vanishing of the Jacobian ideal \(I(f)\), which is generated by \(f\) and the \(n\) derivatives of \(f\). The ideal \(J(f)\) generated by the derivatives alone is the gradient ideal of \(f\). The authors compare the schemes defined by \(I(f)\) and by \(J(f)\) around a singular point \(p\) of \(X\), assuming that \(X\) has only isolated singularities (so that, in particular, it is reduced). The Milnor number of \(X\) at \(p\) is the dimension of the localization \((R/J(f))_p\), while the Tjurina number of \(X\) at \(p\) is defined as the dimension of the localization \((R/I(f))_p\). \(X\) is called `locally Eulerian' if the Milnor number is equal to the Tjurina number at each singular point. The authors prove that \(X\) is locally Eulerian if and only if the symmetric and the Rees algebras of \(I(f)\) in \(R\) are isomorphic. In turn, the condition is equivalent to say that the Tjurina algebra \(R/I(f)\) is artinian Gorenstein. Then, the authors extend their study to the case of reduced projective plane curves \(C\), defining \(C\) to be `of gradient type' if the restriction of \(C\) to each affine chart associated to the singular points is locally Eulerian. The authors prove that curves \(C\) with only simple singularities are of gradient type. Furthermore, they prove that (reduced) quartics are of gradient type, while there are examples of quintics and sextics which are not of gradient type.
Reviewer: Luca Chiantini (Siena)On spectral curves and complexified boundaries of the phase-lock areas in a model of Josephson junction.https://www.zbmath.org/1456.340832021-04-16T16:22:00+00:00"Glutsyuk, A. A."https://www.zbmath.org/authors/?q=ai:glutsyuk.alexey-a"Netay, I. V."https://www.zbmath.org/authors/?q=ai:netay.igor-vA three-parameters family of special double confluent Heun equations, with real parameters \(l\), \(\lambda\), \(\mu\), is considered. The study of the real part of the spectral curve leads to applications to model of Josephson junction which is a family of dynamical systems on 2-torus depending on parameters \((B, A, \omega)\), where \(\omega\) is called the frequency. The authors provide an approach to study the boundaries of the phase-lock areas in \(\mathbb R^2(B, A)\) and their solutions, as \(\omega\) decreases to 0.They prove the irreducibility of the complex spectral curve \(\Gamma_l\) for every \(l\in\mathbb N\). They calculate its genus for \(l\le 20\) and present a conjecture on general genus formula. They apply the irreducibility result to the complexified boundaries of the phase-lock areas of model of Josephson junction. They show as well that its complexification is a complex analytic subset consisting of just four two-dimensional irreducible components, and describe them. They prove that the spectral curve has no real ovals. Finally, they present a Monotonicity Conjecture on the evolution of the phase-lock area portraits, as \(\omega\) decreases, and a partial positive result towards its confirmation.
Reviewer: Bertin Zinsou (Johannesburg)On twists of smooth plane curves.https://www.zbmath.org/1456.111172021-04-16T16:22:00+00:00"Badr, Eslam"https://www.zbmath.org/authors/?q=ai:badr.eslam-e"Bars, Francesc"https://www.zbmath.org/authors/?q=ai:bars.francesc"Lorenzo García, Elisa"https://www.zbmath.org/authors/?q=ai:lorenzo-garcia.elisaLet \(C\) be a projective, smooth, non-hyperelliptic, curve and genus \(g \geq 3\) defined over a field \(k\). Denote by \(\bar{k }\) a fixed separable closure of \(k\) and by \(\bar{C}\) the curve \(C \times_k \bar{k}\). A twist of \(C\) over \(k\) is a projective, non-singular \(C^{\prime}\) defined over \(k\) with a \(\bar{k}\)-isomorphism \(\varphi : \overline{C^{\prime}} \rightarrow \bar{C}\). The paper under review deals with the following question: Assuming that \(C\) admits a smooth \(\bar{k}\)-plane model, does it have a smooth plane model over \(k\)? And if the answer is yes, does every twist of C over
\(k\) also have smooth plane model over \(k\)? The answer, in general, is negative. The twists possessing such models are characterized and an example of a twist not admitting any non-singular plane model over \(k\) is given. An interesting consequence is that explicit equations for a non-trivial Brauer-Severi surface are obtained. Furthermore, for smooth plane curves defined over \(k\) with a cyclic automorphism group generated by a diagonal matrix, a general theoretical result to
compute all its twists is presented.
Reviewer: Dimitros Poulakis (Thessaloniki)Enumerative geometry of plane curves.https://www.zbmath.org/1456.140692021-04-16T16:22:00+00:00"Caporaso, Lucia"https://www.zbmath.org/authors/?q=ai:caporaso.luciaThe author gives a nice account of some results in the enumerative geometry of plane curves. After explaining how curves \(C \subset \mathbb P^2\) of degree \(d\) are parametrized by projective space \(P_d = \mathbb P^{c_d}\) with \(c_d = d(d+3)/2\) via their coefficients,
she gives a very readable proof of the fact that the locus \(S_d \subset P_d\) corresponding to singular curves is a hypersurface of degree \(3(d-1)^2\) in the second section. Then she moves on to consider the more refined \textit{Severi variety} \(S_{d,\delta} \subset P_d\) parametrizing degree \(d\) curves with at least \(\delta\) nodes, explaining why \(S_{d, \delta}\) is empty for \(\delta > \binom{d-1}{2}\). For \(\delta \leq \binom{d-1}{2}\), \textit{J. Harris} [Invent. Math. 84, 445--461 (1986; Zbl 0596.14017)] showed that \(S_{d, \delta}\) is irreducible of dimension \(c_d - \delta\) and the general curve in this family has geometric genus \(g=\binom{d-1}{2} - \delta\). Next the author considers the general problem of computing the number of irreducible plane curves of degree \(d\) and genus \(g\) passing through \(3d+g-1\) general points, explaining the complicated recursive formula of \textit{M. Kontsevich} in the case \(g=0\) [Prog. Math. 129, 335--368 (1995; Zbl 0885.14028)], which originally came from physical considerations as a condition characterizing the associativity of the quantum product on quantum cohomology. For curves of genus \(g>0\), she gives a few examples, but the full story is explained in work of \textit{L. Caporaso} and \textit{J. Harris} [Invent. Math. 131, No. 2, 345--392 (1998; Zbl 0934.14040)].
Reviewer: Scott Nollet (Fort Worth)