On combinatorial bounds for the total Tjurina numbers of certain curves and surfaces with isolated singularities

On com binatorial b ounds for the total Tjurina n um b ers of certain curv es and surfaces with isolated singularities Piotr P okora F ebruary 27, 2026 Abstract W e in vestigate com binatorial b ounds for the total Tjurina num b ers of plane curve arrangemen ts. F o cusing on arrangements of lines and conics in P 2 that admit only ordinary quasi-homogeneous singularities, w e deriv e new structural inequalities gov erning the distribution of multiple intersection p oin ts. As a consequence, w e establish sharp lo w er bounds for the total Tjurina n umbers of free line arrangemen ts with b ounded maximal multiplicit y and, more generally , for free conic-line arrangements. In particular, w e show that for a free arrangement of d lines and k conics, the total Tjurina num b er gro ws at least quadratically in d and k , and w e demonstrate that this b ound is sharp. As an application of these planar results, w e construct a family of surfaces in P 3 with only isolated singularities and arbitrarily large total Tjurina num b ers.This provides new lo w er b ounds for the total Tjurina num b ers of certain h yp ersurfaces that are indep endent of detailed homological data. 1 In tro duction The total Tjurina n umber of a pro jectiv e hypersurface measures the complexit y of its singularities and pla ys a central role in the in terplay betw een singularit y theory , com binatorics, and the homological prop erties of Jacobian algebras. F or plane curv es, a classical result of du Plessis and W all establishes sharp upp er b ounds on the total Tjurina num b er in terms of the degree and the minimal degree of Jacobian relations, with equality c haracterizing free curves [ 6 ]. While these upp er b ounds are rather well understo o d, see for instance [ 2 ], m uch less is kno wn ab out effective low er b ounds, esp ecially in geometrically constrained situations suc h as line or conic-line arrangements with prescrib ed singularities. The aim of this pap er is to establish combinatorial lo wer b ounds for the total Tjurina n um b ers of plane curve arrangements with only ordinary quasi-homogeneous singularities, with a particular fo cus on free arrangemen ts. This w ork con tinues the line of research w e initiated recen tly in [ 14 ], where w e studied the freeness of conic-line arrangemen ts with ordinary quasi- homogeneous singularities, and shifts the fo cus to ward effectiv e lo wer b ounds for their total Tjurina n um b ers. Our results are motiv ated by recen t developmen ts in the theory of plane curv es: b y imposing natural geometric conditions, w e can observe that free arrangements are highly constrained. W e first deriv e structural inequalities for complex line arrangements with b ounded multiplicities, demonstrating that the com binatorics of in tersection p oints strongly restrict p ossible singular configurations. In particular, Theorem 3.5 shows that there 1 Tjurina numbers of v arieties with isola ted singularities P. Pokora are only finitely man y com binatiorial t yp es of supersolv able line arrangements admitting p oin ts of multiplicit y ≤ 4 . W e then consider free conic-line arrangements with ordinary quasi-homogeneous singularities, proving that their total Tjurina num b ers gro w at least quadratically in the num b er of comp onents. This constitutes the main result of the pap er, see Theorem 4.1. A further aim is to extend these planar results to higher dimensions. W e construct surfaces in P 3 with only isolated singularities and arbitrarily large total Tjurina num b ers, obtaining b ounds indep endent of delicate homological data, see Prop osition 5.2. In this wa y , the pap er links the combinatorial geometry of arrangemen ts with the theory of singular hypersurfaces, pro viding new to ols for b ounding and constructing v arieties with con trolled singular b ehavior. 2 Preliminaries Throughout the pap er we w ork ov er the field of complex num b ers C . All v arieties are assumed to b e reduced and p ro jectiv e. W e follow the notation in tro duced in [ 1 ]. Let S = C [ x, y , z ] b e the graded p olynomial ring in three v ariables x, y , z . W e start with general com binatorial preliminaries. Let C ⊂ P 2 b e an arrangemen t of k smo oth curves that admits only ordinary singularities. F or each r ≥ 2 , we denote by n r ( C ) = n r the num b er of ordinary singular p oints of multiplicit y r , i.e., p oints where exactly r curv es from C meet. The maximal m ultiplicit y of the curve C is defined as m ( C ) := max { r : n r ( C )  = 0 } . No w w e recall the basic notions concerning the Tjurina num b er of a plane curve singularity . Let C ⊂ P 2 b e a reduced curve defined by a homogeneous p olynomial f ∈ C [ x, y , z ] , and let p ∈ C b e a singular p oint. Since the problem is local in nature, w e ma y assume (after choosing suitable affine co ordinates) that p = (0 , 0) ∈ C 2 . Th us, in a sufficiently small neigh b orho o d of p , the curve C is giv en b y a con vergen t p o wer series f ( x, y ) ∈ C { x, y } , and we may treat C lo cally as a plane curve in tw o v ariables. Definition 2.1. The Tjurina n umber of C at p = (0 , 0) is defined by τ p ( C ) = dim C C { x, y } ⟨ f , f x , f y ⟩ . The total Tjurina num b er is defined as τ ( C ) = X p ∈ Sing( C ) τ p ( C ) . If p ∈ Sing ( C ) is an ordinary quasi-homogeneous singularity of m ultiplicity r , then τ p ( C ) = ( r − 1) 2 , where r denotes the multiplicit y of C , i.e., the n umber of branc hes passing through that p oin t. Hence, if C ⊂ P 2 is a reduced plane curv e admitting only ordinary quasi-homogeneous 2 Tjurina numbers of v arieties with isola ted singularities P. Pokora singularities, then its total Tjurina num b er is given b y τ ( C ) = X p ∈ Sing( C ) τ p ( C ) = X r ≥ 2 ( r − 1) 2 n r . Finally , let us define the notion of free plane curves. Let C = { f = 0 } ⊂ P 2 b e a reduced plane curve of degree m . Denote b y AR ( f ) := { ( a, b, c ) ∈ S 3 : af x + bf y + cf z = 0 } the mo dule of Jacobian syzygies. Definition 2.2. W e sa y that a reduced plane curve C of degree m is free if AR ( f ) is a free graded S -mo dule of rank 2 . In this case there exist integers ( d 1 , d 2 ) , called the exp onen ts of C , satisfying d 1 + d 2 = m − 1 . F or free curves w e hav e the following fundamen tal c haracterization. Let us define the minimal degree of Jacobian relations of C = { f = 0 } as mdr( f ) := min { r : AR( f ) r  = 0 } . Theorem 2.3 (du Plessis – W all, [ 6 ]) . L et C = { f = 0 } ⊂ P 2 b e a r e duc e d curve of de gr e e m and d 1 = mdr ( f ) . L et us denote the total Tjurina numb er of C by τ ( C ) . Then the fol lowing two c ases hold. a) If d 1 < m/ 2 , then τ ( C ) ≤ τ max ( m, d 1 ) = ( m − 1)( m − d 1 − 1) + d 2 1 and the e quality holds if and only if the curve C is fr e e. b) If m/ 2 ≤ d 1 ≤ m − 1 , then τ ( C ) ≤ τ max ( m, d 1 ) , wher e, in this c ase, we set τ max ( m, d 1 ) = ( m − 1)( m − d 1 − 1) + d 2 1 − 2 m 1 − m + 2 2 ! . 3 Results concerning complex line arrangemen ts with small m ultiplicities Our first result is a structural statement concerning the intersection prop erties of complex line arrangemen ts. It w as originally prov ed in the dual setting of r -ric h lines [ 13 , Prop osition 31]. Ho wev er, w e presen t it here in a form adapted to the framework of this pap er and provide a brief pro of. Prop osition 3.1. L et L ⊂ P 2 b e an arr angement of d ≥ 6 lines such that n r = 0 for al l r > 2 3 d . Then one has n 2 + n 3 + n 4 ≥ d ( d + 15) 18 . Pr o of. W e b egin with the observ ation that, in the presen t setting, the following Hirzebruc h- t yp e inequality holds (see [10, Section 11] or [12, Remark 2.4]): n 2 + 3 4 n 3 ≥ d + X r ≥ 5 r ( r − 4) 2 n r . (1) 3 Tjurina numbers of v arieties with isola ted singularities P. Pokora Next, for every r ≥ 5 we hav e r ( r − 4) 2 ≥ 1 8 · r ( r − 1) 2 . Let us lo ok at the naïve combinatorial coun t in the following form: d 2 ! − n 2 − 3 n 3 − 6 n 4 = X r ≥ 5 r 2 ! n r . Com bining this iden tit y with the Hirzebruch-t yp e inequalit y (1), we obtain n 2 + 3 4 n 3 ≥ d + X r ≥ 5 r ( r − 4) 2 n r ≥ d + 1 8  d 2 ! − n 2 − 3 n 3 − 6 n 4  , hence 9 8 ( n 2 + n 3 + n 4 ) ≥ 9 8 n 2 + 9 8 n 3 + 6 8 n 4 ≥ d ( d + 15) 16 , whic h completes the pro of. No w w e present an interesting application of the ab o ve result in the context of free line arrangemen ts with small m ultiplicities. The first result is general and it w orks without the freeness assumption. Corollary 3.2. L et L ⊂ P 2 b e an arr angement of d ≥ 6 lines such that m ( L ) = 4 . Then n 3 + 3 n 4 ≤ 4 d ( d − 3) 15 . Pr o of. W e start with the naïv e com binatorial coun t: d ( d − 1) 2 = n 2 + n 3 + n 4 + 2 n 3 + 5 n 4 ≥ n 2 + n 3 + n 4 + 5 3 ( n 3 + 3 n 4 ) ≥ d ( d + 15) 18 + 5 3 ( n 3 + 3 n 4 ) , hence we get 5 3 ( n 3 + 3 n 4 ) ≤ 8 d 2 − 24 d 18 , and this completes the pro of. Prop osition 3.3. L et L ⊂ P 2 b e a fr e e arr angement of d ≥ 3 lines such that m ( L ) = 4 . Then ( d − 1)( d − 3) 4 ≤ n 3 + 3 n 4 Pr o of. Recall that [ 9 , Prop osition 3.1] tells us that for free line arrangement with m ( L ) = 4 one has n 2 + n 3 ≤ 3( d − 1) 2 . Then d ( d − 1) 2 = n 2 + n 3 + 2( n 2 + 3 n 3 ) ≤ 3( d − 1) 2 + 2( n 3 + 3 n 4 ) , and the result follows. 4 Tjurina numbers of v arieties with isola ted singularities P. Pokora R emark 3.4 . The ab o v e considerations show that if L ⊂ P 2 is a free arrangement of d ≥ 6 lines with m ( L ) = 4 , then ( d − 1)( d − 3) 4 ≤ n 3 + 3 n 4 ≤ 4 d ( d − 3) 15 . Both b ounds are v ery tigh t. Indeed, consider the simplicial line arrangement A 1 (9) , which consists of d = 9 lines and it has n 2 = 6 , n 3 = 4 , and n 4 = 3 . W e know that A 1 (9) is free. In this case, 12 = (9 − 1)(9 − 3) 4 ≤ n 3 + 3 n 4 = 13 . The upp er b ound is also quite accurate. F or the Hesse arrangemen t of 12 lines with n 2 = 12 and n 4 = 4 , we obtain 27 ≤  4 d ( d − 3) 15  = 28 . These examples suggest that, in contrast to the real case studied in [ 9 ], the family of free complex line arrangements with m ( L ) = 4 might not b e bounded. T o conclude this section, we presen t a result that further narro w the p ossible combinatorics of free arrangements of lines L with m ( L ) = 4 . Recall that an arrangemen t L ⊂ P 2 is called sup ersolv able if it admits a mo dular in tersection point. It is well-kno wn that ev ery sup ersolv able arrangement is free [8]. Theorem 3.5. If L is sup ersolvable arr angement of d ≥ 6 lines with m ( L ) = 4 , then d ∈ { 6 , 7 , 8 , 9 , 10 } . Pr o of. Let us observe that b y [ 3 , Theorem 1.12(1)] a sup ersolv able arrangement L with m ( L ) = 4 satisfies d 1 = 3 . In this case, we ha ve τ ( L ) = ( d − 1) 2 − 3( d − 4) . On the other hand, a straightforw ard combinatorial count gives τ ( L ) = n 2 + 4 n 3 + 9 n 4 = d ( d − 1) 2 + n 3 + 3 n 4 . Comparing these tw o expressions for τ ( L ) , we obtain n 3 + 3 n 4 = d 2 − 9 d + 26 2 . Finally , Corollary 3.2 yields the inequality n 3 + 3 n 4 ≤ 4 d ( d − 3) 15 . Com bining the ab ov e relations sho ws that d ∈ { 6 , 7 , 8 , 9 , 10 } , which completes the pro of. Corollary 3.6. Ther e ar e only finitely many c ombinatorial typ es of sup ersolvable line ar- r angements L with m ( L ) = 4 . 5 Tjurina numbers of v arieties with isola ted singularities P. Pokora 4 A lo w er b ound on the total Tjurina n um b ers of some conic- line arrangements Here is our main result of the pap er. Theorem 4.1. L et C L ⊂ P 2 b e a fr e e arr angement of d lines and k c onics admitting only or dinary quasi-homo gene ous singularities. Then τ ( C L ) ≥ 3 k ( k − 1) + 3 k d + 3( d − 1) 2 4 . Mor e over, this b ound is sharp. Pr o of. Let deg ( C L ) = 2 k + d and d 1 = mdr ( C L ) . The freeness of C L ⊂ P 2 implies that the follo wing equalit y holds: d 2 1 − d 1 (2 k + d − 1) + (2 k + d − 1) 2 = τ ( L ) = X r ≥ 2 ( r − 1) 2 n r . (2) Recall that the following naïv e combinatorial count holds: 4 k 2 ! + 2 k d + d 2 ! = X r ≥ 2 r 2 ! n r , (3) whic h can b e written as (2 k + d − 1) 2 + d − 1 = f 2 − f 1 (4) where f i = P r ≥ 2 r i n r . Let us come back to (2). W e can rewrite it as d 2 1 + d 1 (2 k + d − 1) + (2 k + d − 1) 2 = f 2 − 2 f 1 + f 0 = (2 k + d − 1) 2 + d − 1 − f 1 + f 0 and hence we arrive at d 2 1 − d 1 (2 k + d − 1) +  X r ≥ 2 ( r − 1) n r − d + 1  = 0 . W e compute the discriminan t of the ab ov e equation in d 1 : △ d 1 := (2 k + d − 1) 2 − 4(1 − d ) − 4 X r ≥ 2 ( r − 1) n r . (5) Since C L is free we hav e △ d 1 ≥ 0 , and this giv es us (2 k + d ) 2 + 2 d − 4 k − 3 ≥ X r ≥ 2 (4 r − 4) n r = (2 k + d ) 2 − 4 k − d − X r ≥ 2 ( r 2 − 5 r + 4) n r . Hence 3 2 ( d − 1) + 1 2 X r ≥ 5 ( r 2 − 5 r + 4) n r ≥ n 2 + n 3 . (6) Let us fo cus again on (3), w e hav e 4 k ( k − 1) + 4 k d + d ( d − 1) 2 = X r ≥ 2 r ( r − 1) 2 n r = n 2 + n 3 + 2 n 3 + X r ≥ 4 r ( r − 1) 2 n r (6) ≤ 6 Tjurina numbers of v arieties with isola ted singularities P. Pokora 3 2 ( d − 1) + 2 n 3 + X r ≥ 4 r 2 − 5 r + 4 2 n r + X r ≥ 4 r ( r − 1) 2 n r = 3 2 ( d − 1) + 2 n 3 + X r ≥ 4 ( r 2 − 3 r + 2) n r , hence 4 k ( k − 1) + 4 k d + d 2 − 4 d + 3 4 ≤ n 3 + 1 2 X r ≥ 4 ( r 2 − 3 r + 2) n r . (7) W e now turn to estimating the total Tjurina n um b er of C L . W e hav e τ ( L ) = X r ≥ 2 ( r − 1) 2 n r = X r ≥ 2 r ( r − 1) 2 n r + X r ≥ 2 r 2 − 3 r + 2 2 n r (7) ≥ 4 k ( k − 1) + 4 k d + d ( d − 1) 2 + 4 k ( k − 1) + 4 dk + d 2 − 4 d + 3 4 = 3 k ( k − 1) + 3 k d + 3( d − 1) 2 4 . In order to see that this b ound is sharp, let us recall that there exists a unique (up to pro jectiv e equiv alence) arrangement C L 5 of d = 3 lines and k = 1 conic suc h that n 3 = 3 , see [4, Example 4.14]. In this case w e ha ve τ ( C L 5 ) = 12 , and 3 k ( k − 1) + 3 k d + 3( d − 1) 2 4 = 0 + 9 + 3 · 4 4 = 12 . Hence we obtain equality , which shows that the b ound is sharp. Corollary 4.2. If L ⊂ P 2 is a fr e e arr angement of d lines. Then τ ( L ) ≥ 3( d − 1) 2 4 . The obtained low er b ound is quite accurate, as the following example shows. Let us recall that the Klein arrangemen t K of 21 lines is a free line arrangemen t such that n 3 = 28 and n 4 = 21 , see [11, Chapter 6]. Example 4.3. F or d = 21 we ha v e we τ ( L ) ≥ 300 . W e can easily compute that for the Klein arrangemen t K of 21 lines we hav e τ ( K ) = 301 . Based on the ab o ve considerations, we can formulate the following difficult problem. Pr oblem 4.4 . Classify geometrically free line arrangements L ⊂ P 2 with fixed degree d = |L| and fixed maximal multiplicit y m ( L ) . In particular, determine all p ossible weak combinatorial t yp es and the corresp onding Tjurina num b ers τ ( L ) under these constraints. As a concrete example, consider the case d = 21 with m ( L ) = 4 . It is known that τ ( L ) ∈ { 300 , 301 } , and that there exist 21 weak combinatorial types of such arrangements. Among these is the Klein arrangement. A natural question therefore arises: which of the remaining combinatorial t yp es, aside from the Klein combinatorics, are geometrically realizable? 7 Tjurina numbers of v arieties with isola ted singularities P. Pokora Corollary 4.5. L et L ⊂ P 2 b e a fr e e arr angement of d lines such that n d = 0 with exp onents ( d 1 , d 2 ) . Then d 1 d 2 ≤ ( d − 1) 2 4 . Mor e over, this b ound is sharp if only d 1 = d 2 . Pr o of. Since L is free we get ( d − 1) 2 = d 1 d 2 + τ ( L ) ≥ d 1 d 2 + 3 4 ( d − 1) 2 . Moreo ver, if d 1 = d 2 , then 2 d 1 = d − 1 , and the result follo ws. Example 4.6. The dual Hesse arrangemen t consists of d = 9 lines with n 3 = 12 triple p oints. It is free with exp onen ts ( d 1 , d 2 ) = (4 , 4) . Moreo v er, d 1 d 2 = 4 · 4 = 16 and ( d − 1) 2 4 = 8 2 4 = 16 . Hence d 1 d 2 = ( d − 1) 2 4 , so the b ound is sharp. 5 An application to w ards surfaces with only isolated singular- ities W e apply our b ound on the total Tjurina n umber of conic-line arrangemen ts with quasi- homogeneous singularities from Theorem 4.1 to construct s urfaces in P 3 with only isolated singularities and arbitrarily large total Tjurina num b ers. Let us denote by R = C [ x, y , z , w ] the graded ring of p olynomials in four v ariables. Recall that for a reduced surface X = { F ( x, y , z , w ) = 0 } ⊆ P 3 w e define d ′ 1 = mdr( F ) = min { r : AR ( F ) r  = 0 } , where AR ( F ) = { ( a 1 , a 2 , a 3 , a 4 ) ∈ R 4 : a 1 ∂ x F + a 2 ∂ y F + a 3 ∂ z F + a 4 ∂ w F = 0 } . W e hav e the following general result due to du Plessis and W all [7, Theorem 5.4]. Theorem 5.1. If X = { F ( x, y , z , w ) = 0 } ⊂ P 3 is a r e duc e d surfac e of de gr e e d with at most isolate d singularities, the ( d − 1) 3 − d ′ 1 ( d − 1) 2 ≤ τ ( X ) ≤ ( d − 1) 3 − d ′ 1 ( d − d ′ 1 − 1)( d − 1) . In light of the ab o v e result, we present our contribution to ward b ounding the total Tjurina n umber of certain surfaces that is in the spirit of [ 5 , Remark 3.2]. The main adv an tage of our result is that the b ound do es not dep end explicitly on the homological data associated with the surface. Prop osition 5.2. L et C = { f ( x, y , z ) = 0 } ⊂ P 2 b e an arr angement of k c onics and d lines with only or dinary quasi-homo gene ous singularities. A ssume that C is fr e e. Consider the surfac e X = { F ( x, y , z , w ) = f ( x, y , z ) + w 2 k + d = 0 } ⊂ P 3 . 8 Tjurina numbers of v arieties with isola ted singularities P. Pokora Then X has only isolate d singularities and τ ( X ) ≥ (2 k + d − 1) ·  3 k ( k − 1) + 3 k d + 3( d − 1) 2 4  . Pr o of. The fact that X has only isolated singularities follows from straigh tforward computa- tions of its partial deriv ativ es. T o obtain a low er b ound on τ ( X ) , we note that the problem is lo cal. Hence, w e p erform computations around each singular p oint, and we ha ve a naïve Thom-Sebastiani principle for isolated singularities, namely τ  f ( x, y , z ) + w 2 k + d  = τ ( f ( x, y , z )) · τ ( w 2 k + d ) = (2 k + d − 1) · τ ( f ( x, y , z )) , whic h completes the pro of. F unding I would like to warmly thank Alex Dimca for all comments regarding the con tent of the pap er and for suggesting Theorem 3.5. Piotr P ok ora are supp orted by the National Science Cen tre (Poland) Sonata Bis Grant 2023/50/E/ST1/00025 . F or the purp ose of Op en A ccess, the author has applied a CC-BY public cop yrigh t license to an y A uthor A ccepted Man uscript (AAM) version arising from this submission. References [1] A. Dimca, Hyp erplane arr angements. A n intr o duction . Universitext. 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Pok ora, The orbifold Langer-Miyaoka-Y au inequality and Hirzebruch-t yp e inequalities. Ele ctr on. R es. A nnounc. Math. Sci. 24 : 21 – 27 (2017). [13] P . P okora, Hirzebruch-t yp e inequalities viewed as to ols in combinatorics. Ele ctr on. J. Comb. 28(1) : Researc h Paper P1.9, 22 p. (2021). [14] P . P okora, On Poincaré p olynomials for plane curves with quasi-homogeneous singularities. Bul l. L ond. Math. So c. 57(8) : 2549 – 2560 (2025). Piotr Pok ora Departmen t of Mathematics, Univ ersity of the National Education Commission Krako w, P o dc hor ¸ ażyc h 2, PL-30-084 Krakó w, Poland. Email: piotr.pokora@uken.krakow.pl 10