Nicola Pace

Bio: Nicola Pace is an academic researcher from Technische Universität München. The author has contributed to research in topics: Projective plane & Complete graph. The author has an hindex of 7, co-authored 20 publications receiving 109 citations. Previous affiliations of Nicola Pace include Florida Atlantic University & University of São Paulo.

3-Nets realizing a group in a projective plane

Abstract: In a projective plane $\mathit{PG}(2,\mathbb{K})$ defined over an algebraically closed field $\mathbb{K}$ of characteristic 0, we give a complete classification of 3-nets realizing a finite group. An infinite family, due to Yuzvinsky (Compos. Math. 140:1614---1624, 2004), arises from plane cubics and comprises 3-nets realizing cyclic and direct products of two cyclic groups. Another known infinite family, due to Pereira and Yuzvinsky (Adv. Math. 219:672---688, 2008), comprises 3-nets realizing dihedral groups. We prove that there is no further infinite family. Urzua's 3-nets (Adv. Geom. 10:287---310, 2010) realizing the quaternion group of order 8 are the unique sporadic examples. If p is larger than the order of the group, the above classification holds in characteristic p>0 apart from three possible exceptions $\rm{Alt}_{4}$ , $\rm{Sym}_{4}$ , and $\rm{Alt}_{5}$ . Motivation for the study of finite 3-nets in the complex plane comes from the study of complex line arrangements and from resonance theory; see (Falk and Yuzvinsky in Compos. Math. 143:1069---1088, 2007; Miguel and Buzunariz in Graphs Comb. 25:469---488, 2009; Pereira and Yuzvinsky in Adv. Math. 219:672---688, 2008; Yuzvinsky in Compos. Math. 140:1614---1624, 2004; Yuzvinsky in Proc. Am. Math. Soc. 137:1641---1648, 2009).

k-nets embedded in a projective plane over a field

Abstract: We investigate k-nets with k?4 embedded in the projective plane PG(2, $\mathbb{K}$ ) defined over a field $\mathbb{K}$ ; they are line configurations in PG(2, $\mathbb{K}$ ) consisting of k pairwise disjoint line-sets, called components, such that any two lines from distinct families are concurrent with exactly one line from each component. The size of each component of a k-net is the same, the order of the k-net. If $\mathbb{K}$ has zero characteristic, no embedded k-net for k?5 exists; see [11,14]. Here we prove that this holds true in positive characteristic p as long as p is sufficiently large compared with the order of the k-net. Our approach, different from that used in [11,14], also provides a new proof in characteristic zero.

3-nets realizing a group in a projective plane

Abstract: In a projective plane PG(2,K) defined over an algebraically closed field K of characteristic 0, we give a complete classification of 3-nets realizing a finite group. An infinite family, due to Yuzvinsky, arises from plane cubics and comprises 3-nets realizing cyclic and direct products of two cyclic groups. Another known infinite family, due to Pereira and Yuzvinsky, comprises 3-nets realizing dihedral groups. We prove that there is no further infinite family. Urzua's 3-nets realizing the quaternion group of order 8 are the unique sporadic examples. If p is larger than the order of the group, the above classification holds true in characteristic p>0 apart from three possible exceptions Alt_4, Sym_4 and Alt_5.

On Small Complete Arcs and Transitive A5‐Invariant Arcs in the Projective Plane PG(2,q)

Abstract: Let q be an odd prime power such that q is a power of 5 or (mod 10). In this case, the projective plane admits a collineation group G isomorphic to the alternating group A5. Transitive G-invariant 30-arcs are shown to exist for every . The completeness is also investigated, and complete 30-arcs are found for . Surprisingly, they are the smallest known complete arcs in the planes , and . Moreover, computational results are presented for the cases and . New upper bounds on the size of the smallest complete arc are obtained for .

Infinite family of large complete arcs in PG(2, qn), with q odd and n > 1 odd

Abstract: For q odd and n > 1 odd, a new infinite family of large complete arcs K? in PG(2, q n ) is constructed from complete arcs K in PG(2, q) which have the following property with respect to an irreducible conic $${\mathcal{C}}$$ in PG(2, q): all the points of K not in $${\mathcal{C}}$$ are all internal or all external points to $${\mathcal{C}}$$ according as q ? 1 (mod 4) or q ? 3 (mod 4).

A first course in coding theory, by Raymond Hill. Pp 251. £12·50. 1986. ISBN 0-19-853803-0 (Oxford University Press)

Abstract: Algebraic coding theory is a new and rapidly developing subject, popular for its many practical applications and for its fascinatingly rich mathematical structure. This book provides an elementary yet rigorous introduction to the theory of error-correcting codes. Based on courses given by the author over several years to advanced undergraduates and first-year graduated students, this guide includes a large number of exercises, all with solutions, making the book highly suitable for individual study.

3-Nets realizing a group in a projective plane

Abstract: In a projective plane $\mathit{PG}(2,\mathbb{K})$ defined over an algebraically closed field $\mathbb{K}$ of characteristic 0, we give a complete classification of 3-nets realizing a finite group. An infinite family, due to Yuzvinsky (Compos. Math. 140:1614---1624, 2004), arises from plane cubics and comprises 3-nets realizing cyclic and direct products of two cyclic groups. Another known infinite family, due to Pereira and Yuzvinsky (Adv. Math. 219:672---688, 2008), comprises 3-nets realizing dihedral groups. We prove that there is no further infinite family. Urzua's 3-nets (Adv. Geom. 10:287---310, 2010) realizing the quaternion group of order 8 are the unique sporadic examples. If p is larger than the order of the group, the above classification holds in characteristic p>0 apart from three possible exceptions $\rm{Alt}_{4}$ , $\rm{Sym}_{4}$ , and $\rm{Alt}_{5}$ . Motivation for the study of finite 3-nets in the complex plane comes from the study of complex line arrangements and from resonance theory; see (Falk and Yuzvinsky in Compos. Math. 143:1069---1088, 2007; Miguel and Buzunariz in Graphs Comb. 25:469---488, 2009; Pereira and Yuzvinsky in Adv. Math. 219:672---688, 2008; Yuzvinsky in Compos. Math. 140:1614---1624, 2004; Yuzvinsky in Proc. Am. Math. Soc. 137:1641---1648, 2009).

k-nets embedded in a projective plane over a field

Abstract: We investigate k-nets with k?4 embedded in the projective plane PG(2, $\mathbb{K}$ ) defined over a field $\mathbb{K}$ ; they are line configurations in PG(2, $\mathbb{K}$ ) consisting of k pairwise disjoint line-sets, called components, such that any two lines from distinct families are concurrent with exactly one line from each component. The size of each component of a k-net is the same, the order of the k-net. If $\mathbb{K}$ has zero characteristic, no embedded k-net for k?5 exists; see [11,14]. Here we prove that this holds true in positive characteristic p as long as p is sufficiently large compared with the order of the k-net. Our approach, different from that used in [11,14], also provides a new proof in characteristic zero.