A linear system of plane curves satisfying multiplicity conditions at points in general position is called special if the dimension is larger than the expected dimension. A (-1) curve is an irreducible curve with self intersection -1 and genus zero. The Harbourne-Hirschowitz Conjecture is that a linear system is special only if a multiple of some fixed (-1) curve is contained in every curve of the linear system. This conjecture is proven for linear systems with multiplicity four at all but one of the points.
In this paper we prove the Harbourne-Hirschowitz conjecture for quasi-homogeneous linear systems of multiplicity 6 on P^2. For the proof we use the degeneration of the plane by Ciliberto and Miranda and results by Laface, Seibert, Ugaglia and Yang. As an application we derive a classification of the special systems of multiplicity 6.
In this paper we consider linear systems of $\mathbb{P}^2$ with all but one of the base points of multiplicity 5. We give an explicit way to evaluate the dimensions of such systems.
In the paper we prove Harbourne-Hirschowitz conjecture for quasi-homogeneous linear systems on $\mathbb P^2$ for $m=7$, 8, 9, 10, i.e. systems of curves of given degree passing through points in general position with multiplicities at least $m,...,m,m_0$, where $m=7$, 8, 9, 10, $m_0$ is arbitrary.
The Segre-Gimigliano-Harbourne-Hirschowitz Conjecture can be naturally formulated for Hirzebruch surfaces F_n. We show that this Conjecture holds for imposed base points of equal multiplicity bounded by 8.
In this paper we prove that for all pairs $(d,m)$ with $d/m \geq 174/55$, the linear system of plane curves of degree $d$ with ten general base points of multiplicity $m$ has the expected dimension.
Let X=G/B be a complete flag variety, and L' and L" two line bundles on X. Consider the cup product map H^{d'}(X,L') x H^{d"}(X, L") --> H^{d}(X,L), where L=L' x L" and d=d'+d". We answer two natural questions about the map above: When is it a nonzero map of irreducible G-representations? Conversely, given generic irreducible representations V' and V" of G, which irreducible components of V' x V" may appear in the right hand side of the map above? We also give bounds on the multiplicities appearing in a tensor product, and relate these considerations to the boundary of the Littlewood-Richardso…
We prove: if $d/m < 2280/721$, there is no curve of degree $d$ passing through $n = 10$ general points with multiplicity $m$ in $\bf{P}^2$. Similar results are given for other special values of $n$. Our bounds can be naturally written as certain palindromic continued fractions.
We extend the range of linear systems of plane curves for which it is known that the dimension is equal to the expected dimension. We use a degeneration technique due to C. Ciliberto and R. Miranda. From our result we deduce, as a Corollary, a theorem due to Evain on homogeneous systems of curves through a number of base points given by a power of 4.
In this paper we give a new proof of the fact that for all pairs of positive integers (d, m) with d/m < 117/37, the linear system of plane curves of degree d with ten general base points of multiplicity m is empty.
We classify all momentum polytopes of rank one for multiplicity free quasi-Hamiltonian $K$-manifolds for simple and simply connected Lie groups $K$ by using the methods developed in a recent paper by F. Knop. This leads to lots of new concrete examples of multiplicity free quasi-Hamiltonian manifolds or equivalently, Hamiltonian loop group actions.
We show the possible Milnor numbers of deformations of semi-quasi-homogeneous isolated plane curve singularities. In Theorem 1.1 we list integers can be attained as Milnor numbers of a given semi-quasi-homogeneous singularity.
Inspired by the work of Bhatt and Singh (see: arXiv:1307.1171) we compute the $F$-pure threshold of quasi-homogeneous polynomials. We first consider the case of a curve given by a quasi-homogeneous polynomial $f$ in three variables $x,y,z$ of degree equal to the degree of $xyz$ and then we proceed with the general case of a Calabi-Yau hypersurface, i.e. a hypersurface given by a quasi-homogeneous polynomial $f$ in $n+1$ variables $x_0, \ldots, x_n$ of degree equal to the degree of $x_0 \cdots x_n$.
We find explicit free resolutions for the $\scr D$-modules ${\scr D} f^s$ and ${\scr D}[s] f^s/{\scr D}[s] f^{s+1}$, where $f$ is a reduced equation of a locally quasi-homogeneous free divisor. These results are based on the fact that every locally quasi-homogeneous free divisor is Koszul free, which is also proved in this paper
Starting from a cubic form, we give a general construction of a quasi-complete homogeneous manifold endowed with a natural contact structure. We show that it can be compactified into a projective contact manifold if and only if the cubic form is the determinant of a simple cubic Jordan algebra.
The aim of this paper is to show the possible Milnor numbers of deformations of semi-quasi-homogeneous isolated plane curve singularities. Main result states that if $f$ is irreducible and nondegenerate, by deforming $f$ one can attain all Milnor numbers ranging from $μ(f)$ to $μ(f)-r(p-r)$, where $r$ and $p$ are easily computed from the Newton diagram of $f$.
We show that the Hilbert-Kunz multiplicity is a rational number for an R_+-primary homogeneous ideal I=(f_1, ..., f_n) in a two-dimensional graded domain R of finite type over an algebraically closed field of positive characteristic. Moreover we give a formula for the Hilbert-Kunz multiplicity in terms of certain rational numbers coming from the strong Harder-Narasimhan filtration of the syzygy bundle Syz(f_1, . . ., f_n) on the projective curve Y = Proj R.
Let $R$ be a polynomial ring and $I \subset R$ be a perfect ideal of height two minimally generated by forms of the same degree. We provide a formula for the multiplicity of the saturated special fiber ring of $I$. Interestingly, this formula is equal to an elementary symmetric polynomial in terms of the degrees of the syzygies of $I$. Applying ideas introduced in arXiv:1805.05180, we obtain the value of the j-multiplicity of $I$ and an effective method for determining the degree and birationality of rational maps defined by homogeneous generators of $I$.
Let Z be a fat point scheme in P^2 supported on general points. Here we prove that if the multiplicities are at most 3 and the length of Z is sufficiently high then the number of generators of the homogeneous ideal I_Z in each degree is as small as numerically possible. Since it is known that Z has maximal Hilbert function, this implies that Z has the expected minimal free resolution.
Let $R$ be a polynomial ring over a field and $I \subset R$ be a Gorenstein ideal of height three that is minimally generated by homogeneous polynomials of the same degree. We compute the multiplicity of the saturated special fiber ring of $I$. The obtained formula depends only on the number of variables of $R$, the minimal number of generators of $I$, and the degree of the syzygies of $I$. Applying results from arXiv:1805.05180, we get a formula for the $j$-multiplicity of $I$ and an effective method to study a rational map determined by a minimal set of generators of $I$.
This paper concerns the question of whether a more direct limit can be used to obtain the limit Hilbert-Kunz multiplicity, a possible candidate for a characteristic zero Hilbert-Kunz multiplicity. The main goal is to establish an affirmative answer for one of the main cases for which the limit Hilbert-Kunz multiplicity is even known to exist, namely that of graded ideals in the homogeneous coordinate ring of smooth projective curves. The proof involves more careful estimates of bounds found independently by Brenner and Trivedi on the dimensions of the cohomologies of twists of the syzygy bundl…
We describe all connected components of the space of hyperbolic Gorenstein quasi-homogeneous surface singularities. We prove that any connected component is homeomorphic to a quotient of R^d by a discrete group.
Let K be a simple and simply connected compact Lie group. We call a (twisted) quasi-Hamiltonian K-manifold M a quasi-Hamiltonian model space if it is multiplicity free and its momentum map is surjective. We explicitly identify the subgroups of the Lie algebra of the maximal torus of K, which, by F. Knop's classification of multiplicity free quasi-Hamiltonian manifolds, are in one-to-one correspondence with the isomorphism classes of quasi-Hamiltonian model K-spaces.
Given r>=n quasi-homogeneous polynomials in n variables, the existence of a certain duality is shown and explicited in terms of generalized Morley forms. This result, that can be seen as a generalization of [3,corollary 3.6.1.4] (where this duality is proved in the case r=n), was observed by the author at the same time. We will actually closely follow the proof of (loc. cit.) in this paper.
In this work, we consider a quasi-homogeneous, corank $1$, finitely determined map germ $f$ from $(\mathbb{C}^2,0)$ to $(\mathbb{C}^3,0)$. We consider the invariants $m(f(D(f))$ and $J$, where $m(f(D(f))$ denotes the multiplicity of the image of the double point curve $D(f)$ of $f$ and $J$ denotes the number of tacnodes that appears in a stabilization of the transversal slice curve of $f(\mathbb{C}^2)$. We present formulas to calculate $m(f(D(f))$ and $J$ in terms of the weights and degrees of $f$.
