1,721,027 research outputs found
Special effect varieties and (-1)-curves
No full textHere we introduce the concept of special effect curve which permits to study, from a different point of view, special linear systems in P^2, i.e., linear system with general multiple base points whose effective dimension is strictly greater than the expected one. In particular we study two different kinds of special effect: the α-special effect is defined by requiring some numerical conditions, while the definition of h1-special effect concerns cohomology groups. We state two new conjectures for the characterization of special linear systems and we prove they are equivalent to the Segre and the Harbourne-Hirschowitz ones
Comparing powers and symbolic powers of ideals
No full textWe develop tools to study the problem of containment of symbolic powers I^(m) in powers I^r for a homogeneous ideal I in a polynomial ring k[PN] in N + 1 variables over an arbitrary algebraically closed field k. We obtain results on the structure of the set of pairs (r, m) such that I^(m) is contained in I^r. As corollaries, we show that I^2 contains I^(3) whenever S is a finite generic set of points in P^2 (thereby giving a partial answer to a question of Huneke), and we show that the containment theorems of Ein-Lazarsfeld-Smith [Invent. Math. 144 (2001), pp. 241-252] and Hochster-Huneke [Invent. Math. 147 (2002), pp. 349-369] are optimal for every fixed dimension and codimension
On the identifiability of binary Segre products
No full textWe prove that a product of m > 5 copies of P^1, embedded in the projective space P^r by the standard Segre embedding, is k-identifiable (i.e. a general point of the secant variety S^k(X) is contained in only one (k + 1)-secant k-space), for all k such that k + 1 ≤ 2m−1/m. The research that led to the present paper was partially supported by a grant of the group GNSAGA of INdAM
The cohomology of rank 2 bundles on P^2
No full textWe classify all sequences of integers that can be, up to a shift, the cohomology sequence {h 1 (E(n))} of a rank 2 bundle E on P^2. We show how some of the main invariants of the bundle can be read from the sequence
The resurgence of ideals of points and the containment problem
No full textWe relate properties of linear systems on X to the question of when I^r contains I^(m) in the case that I is the homogeneous ideal of a finite set of distinct points p_1,...,p_n in P^2, where X is the surface obtained by blowing up the points. We obtain complete answers for when I^r contains I^(m) when the points p_i's lie on a smooth conic or when the points are general and n ≤ 9
Dall'eredità grassmanniana alla teoria delle omografie nella scuola di Peano
No full textIn questo lavoro presentiamo una ricostruzione ed un’analisi storica del processo teorico che nell’ambito della scuola di Peano portò dall’eredità grassmanniana al calcolo vettoriale e alla teoria delle omografie. Il nostro obiettivo è anche tentare di dare una generalizzazione delle idee basilari introdotte da Peano (e da H.Grassmann). Inoltre analizziamo le applicazioni del calcolo geometrico Peano alle dimostrazioni di fondamentali teoremi di geometria proiettiva. Infine viene esaminata analiticamente un’importante applicazione fisico matematica di Roberto Marcolongo della teoria delle omografie per quanto riguarda le trasformazioni di Lorentz
A tropical interpretation of m-dissimilarity maps
No full textLet T be a weighted tree with n numbered leaves and let D = (D (i, j))i, j be its distance matrix, so D (i, j) is the distance between the leaves i and j. If m is an integer satisfying 2 ≤ m ≤ n, we prove a tropical formula to compute the m-dissimilarity map of T (i.e. the weights of the subtrees of T with m leaves), given D. For m = 3, we present a tropical description of the set of m-dissimilarity maps of trees. For m = 4, a partial result is given
Max-plus objects to study the complexity of graphs
No full textGiven an undirected graph G, we define a new object H_G, called the mp-chart of , in the max-plus algebra. We use it, together with the max-plus permanent, to describe the complexity of graphs. We show how to compute the mean and the variance of H_G in terms of the adjacency matrix of G and we give a central limit theorem for H_G. Finally, we show that the
mp-chart is easily tractable also for the complement graph
Catalecticant intersection and confinement of decompositions of forms
Full text linkWe introduce the notion of confinement of decompositions for forms or vector of forms. The confinement, when it holds,
lowers the number of parameters that one needs to consider, in order to find all the possible decompositions of a given set of data.
With the technique of confinement, we obtain here two results.
First, we give a new, shorter proof of a result by London that 3 general plane cubics have 2 simultaneous Waring decompositions of rank 6.
Then we compute, with the software Bertini, that 4 general plane quartics have 18 different decompositions of rank 10 (a result which was not known before)
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