Plane curves with small linear orbits I

The `linear orbit' of a plane curve of degree d is its orbit in the projective space of dimension d(d+3)/2 parametrizing such curves under the natural action of PGL(3). In this paper we compute the degree of the closure of the linear orbits of most curves with positive dimensional stabilizers. Our tool is a nonsingular variety dominating the orbit closure, which we construct by a blow-up sequence mirroring the sequence yielding an embedded resolution of the curve. The results given here will serve as an ingredient in the computation of the analogous information for arbitrary plane curves. Linear orbits of smooth plane curves are studied in [A-F1].Comment: 34 pages, 4 figures, AmS-TeX 2.1, requires xy-pic and eps

Inclusion-exclusion and Segre classes

We propose a variation of the notion of Segre class, by forcing a naive `inclusion-exclusion' principle to hold. The resulting class is computationally tractable, and is closely related to Chern-Schwartz-MacPherson classes. We deduce several general properties of the new class from this relation, and obtain an expression for the Milnor class of a scheme in terms of this class.Comment: 8 page

Verdier specialization via weak factorization

Let X in V be a closed embedding, with V - X nonsingular. We define a constructible function on X, agreeing with Verdier's specialization of the constant function 1 when X is the zero-locus of a function on V. Our definition is given in terms of an embedded resolution of X; the independence on the choice of resolution is obtained as a consequence of the weak factorization theorem of Abramovich et al. The main property of the specialization function is a compatibility with the specialization of the Chern class of the complement V-X. With the definition adopted here, this is an easy consequence of standard intersection theory. It recovers Verdier's result when X is the zero-locus of a function on V. Our definition has a straightforward counterpart in a motivic group. The specialization function and the corresponding Chern class and motivic aspect all have natural `monodromy' decompositions, for for any X in V as above. The definition also yields an expression for Kai Behrend's constructible function when applied to (the singularity subscheme of) the zero-locus of a function on V.Comment: Minor revision. To appear in Arkiv f\"or Matemati

Algebro-geometric Feynman rules

We give a general procedure to construct algebro-geometric Feynman rules, that is, characters of the Connes-Kreimer Hopf algebra of Feynman graphs that factor through a Grothendieck ring of immersed conical varieties, via the class of the complement of the affine graph hypersurface. In particular, this maps to the usual Grothendieck ring of varieties, defining motivic Feynman rules. We also construct an algebro-geometric Feynman rule with values in a polynomial ring, which does not factor through the usual Grothendieck ring, and which is defined in terms of characteristic classes of singular varieties. This invariant recovers, as a special value, the Euler characteristic of the projective graph hypersurface complement. The main result underlying the construction of this invariant is a formula for the characteristic classes of the join of two projective varieties. We discuss the BPHZ renormalization procedure in this algebro-geometric context and some motivic zeta functions arising from the partition functions associated to motivic Feynman rules.Comment: 26 pages, LaTeX, 1 figur

Intraoral transposition of pedicled temporalis muscle flap followed by zygomatic implant placement

Despite the recent advances of sophisticated reconstructive surgical techniques, management of maxillectomy defects continues to be challenging. For a selected group of patients, who cannot sustain a sophisticated microsurgical reconstructive procedure, a prosthetic obturator is indicated to separate the oral cavity from the sinonasal cavities. After the development of the osseointegration concept, dental implants have proven to be indicated for the rehabilitation of patients who underwent maxillectomy. Recently, surgeons can use a computer-assisted software package, which enables them to insert implants after a detailed analysis of the residual bone. For some patients with limited amount of residual maxillary bone, unusual surgical sites such as the zygomatic complex have been tested. We introduce a successful 2-step surgical procedure using a pedicled temporalis muscle flap and zygomatic implant placement to reconstruct a maxillary defect after oncological resection

PROPOSAL OF A PRESURGICAL ALGORITHM FOR PATIENTS AFFECTED BY OBSTRUCTIVE SLEEP APNEA SYNDROME

PURPOSE: To propose an algorithm for the preoperative management of patients with obstructive sleep apnea syndrome (OSAS) and review the surgical outcomes in such patients. MATERIALS AND METHODS: This prospective cohort study involved 71 patients with OSAS who underwent presurgical upper airway endoscopy and cephalometry before being assigned to treatment categories based on the site(s) of obstruction, the pattern of collapse, the characteristics of the soft tissue, the air space between the base of the tongue and the posterior wall of the pharynx, and the severity of OSAS. Six months after surgery, they were followed up using polysomnography and the Epworth Sleepiness Scale. The pre- and postsurgical data were compared using a paired Student t test. RESULTS: The mean preoperative apnea/hypopnea index of the 71 patients (61 male and 10 female) was 40.98 events/hour (range, 14.7 to 87.6 events/hr), and the mean postoperative apnea/hypopnea index was 13.96 events/hour (range, 0 to 20 events/hr). The difference was statistically significant (P < .001). CONCLUSIONS: This algorithm was developed on the principle that every patient with OSAS should be considered individually. In the authors' opinion, taking into account the number, site(s), pattern, and degree of the collapse/obstruction is a reasonable means of ensuring the correct diagnosis and treatment