Extremal Singularities in Positive Characteristic
Zhibek Kadyrsizova, Jennifer Kenkel, Janet Page, Jyoti Singh, Karen E. Smith, Adela Vraciu, Emily E. Witt
aa r X i v : . [ m a t h . A C ] S e p EXTREMAL SINGULARITIES IN POSITIVE CHARACTERISTIC
ZHIBEK KADYRSIZOVA, JENNIFER KENKEL, JANET PAGE, JYOTI SINGH,KAREN E. SMITH, ADELA VRACIU, AND EMILY E. WITT
Abstract.
We prove a general lower bound on the F -pure threshold of a reduced formof characteristic p > in terms of its degree, and investigate the class of forms thatachieve this minimal possible F -pure threshold. Specifically, we prove that if f is areduced homogenous polynomial of degree d , then its F -pure threshold (at the uniquehomogeneous maximal ideal) is at least d − . We show, furthermore, that its F -purethreshold equals d − if and only if f ∈ m [ q ] and d = q + 1 , where q is a power of p . Up tolinear changes of coordinates (over a fixed algebraically closed field), we show that thereare only finitely many such "extremal singularities" of bounded degree and embeddingdimension, and only one with isolated singularity. Finally, we indicate several ways inwhich the projective hypersurfaces defined by such forms are "extremal," for example,in terms of the configurations of lines they can contain. Introduction
Fix an algebraically closed field k of characteristic p > . What is the most singularpossible reduced hypersurface over k ?The multiplicity is the first crude measurement of singularity—roughly speaking highermultiplicity singularities are more singular. But we want to identify which singularitiesare the most singular, even taking multiplicity into account. Among multiplicity twosingularities, for example, the cusp y − x is more singular than the normal crossing xy ,but also the cusp is more singular in some characteristics than others.The F -pure threshold is a more refined numerical invariant for comparing singularities ofhypersurfaces in positive characteristic. Analogous to—but much more subtle than—thelog canonical threshold for a complex hypersurface, the F -pure threshold is equal to oneat each smooth point (or more generally, at each F -pure point), with "worse singularities"having smaller F -pure thresholds. Using the F -pure threshold, can we identify a classof singularities that is "maximally singular"? What are the properties of such "extremalsingularities"?This paper presents a general lower bound on the F -pure threshold of a reduced ho-mogeneous form in terms of its degree—that is, the multiplicity at the origin of thecorresponding affine hypersurface. We show that our lower bound is sharp, enabling us This paper grew out of discussions begun at the AWM-sponsored workshop "Women in CommutativeAlgebra" at the Banff International Research Station on a project proposed and led by the fifth author.Partial funding for participants was supplied by NSF grant numbers 193439 and NSF-HRD 150048. Inaddition, partial funding was provided by SERB(DST) grant number ECR/2017/000963 (for Jyoti Singh),NSF grant number 1801697 (for Karen Smith), and NSF CAREER grant 1945611 (for Emily Witt). o interpret "maximally singular" hypersurfaces as those for which this minimal possible F -pure threshold is achieved. These extremal singularities, as we call them, turn outto have many interesting algebraic and geometric properties which we treat in Sections5 and 8, respectively. These properties are special to prime characteristic: nothing likethis happens for complex singularities, and indeed, there are much larger lower boundsknown for the log canonical threshold of a homogeneous complex singularity in terms ofits degree (see paragraph 1.4).While there has been much research into computing the F -pure threshold in specificsettings (see, for example, [BS15], [Her16], [HNWZ16], [HT17]), we are not aware of anyprior research into lower bounds on the F -pure threshold. Lower bounds on log canonicalthresholds, on the other hand, have been studied in [CP02], [dFEM03], and [DP14], forexample.Our main theorem is the following lower bound on the F -pure threshold: Theorem 1.1.
Fix any field k of positive characteristic p . Let f ∈ k [ x , . . . , x n ] be areduced homogeneous polynomial of degree d = deg( f ) ≥ . Then (1) fpt( f ) ≥ d − . Furthermore, equality holds in (1) if and only if d = q + 1 , where q is a power of p and f ∈ h x q , . . . , x qn i . Theorem 1.1 implies that a homogeneous polynomial for which the F -pure thresholdachieves the lower bound (1) is a Frobenius form , by which we mean a polynomial x q L + · · · + x qn L n , where the L i are linear forms. Frobenius forms should not be confused with the muchmore special type of form in prime characteristic called a Hermitian form; see 1.3.Frobenius forms have a convenient matrix representation, which we exploit in the secondhalf of the paper to examine extremal singularities in detail. For example, we prove: Theorem 1.2.
Fix an algebraically closed ground field k of characteristic p > . Thereare only finitely many isomorphism types of singularities achieving the minimal possible F -pure threshold d − in each degree d and embedding dimension n over k . Indeed, thenumber is bounded above by the n -th Fibonacci number. Only one of these has isolatedsingularities, namely the one represented by the "diagonal" form x q +11 + x q +12 + · · · + x q +1 n ,where q is a power of p . Among degree three singularities of embedding dimension three, for example, we haveextremal singularities only when the characteristic is two. In this case, there is exactly onewith isolated singularity— namely the cone over the super-singular elliptic curve defined(up to linear change of coordinates) by the Frobenius form x + y + z . There are twoothers, each with one-dimensional singular locus, the cusp x + y z and the reducible formdefining a smooth conic with a tangent line. This classification follows from the proof ofTheorem 7.1. heorem 1.2 is a combination of Theorems 6.1 and 7.1 proved in Sections 6 and 7, re-spectively. The isolated singularity case, Theorem 6.1, follows from a theorem of Beauvillewhen there are at least variables, but we give a straightforward new argument. We be-lieve Theorem 1.2 is new when the singular locus has higher dimension. The proof exploitsthe matrix form of a Frobenius form, using quite delicate combinatorics.In Section 8, we point out a few extremal geometric properties of the projective hyper-surfaces defined by Frobenius forms. For example, it is easy to see that they have highlyinseparable Gauss maps and that their dual hypersurfaces (in the smooth case) are alsoextremal (Proposition 8.5). We show also that a projective hypersurface is extremal ifand only if every hyperplane section is extremal (Theorem 8.2), and that their extremalnature is reflected in the very special configurations of lines they can contain (Proposi-tion 8.11). As a special case, we recover the fact that an extremal cubic surface has theproperty that all intersecting lines on it meet at an Eckard point [KKP + Comparison to Hermitian Forms in Prime Characteristic.
A very specialtype of Frobenius form—namely a characteristic p Hermitian form —has long been knownto be extremal with respect to the number of rational points it contains; see, for example,[Seg65], [BC66], [HK16]. The existence of many rational points is a consequence of themany linear spaces contained in the projective hypersurface defined by a Frobenius form;see the discussion in [Kol15, §35] for example. For the definition of Hermitian form, anda discussion of how special they are among Frobenius forms, see Remark 5.6.1.4.
Comparison to Lower Bounds for Log Canonical threshold.
Our lower boundon the F -pure threshold immediately implies an analogous bound on the log canonicalthreshold of a complex hypersurface by reduction to characteristic p ; see §2.3. However,while sharp in prime characteristics, the corresponding bound for the log canonical thresh-old is far from sharp. This is to be expected: singularities in prime characteristic can be"bad" in ways not possible over C .For example, de Fernex, Ein, and Mustaţă [dFEM03] prove that for a complex homoge-neous polynomial of degree d in n variables, the log canonical threshold is bounded belowby min (cid:18) n − rd , (cid:19) , where r is the dimension of the singular locus of the corresponding affine hypersurface.The corresponding statement in positive characteristic, however, is spectacularly false.For example, the polynomial in characteristic p defined by x p e +11 + x p e +12 + · · · + x p e +1 n has F -pure threshold p e (a simple case of Theorem 1.1), which is much smaller, especiallyas n grows large, than np e +1 , the value of the de Fernex-Ein-Mustaţă bound in this case(and indeed the log canonical threshold when n ≤ p e + 1 ). Acknowledgements.
The authors thank Karl Schwede, Mircea Mustaţă, and JánosKollár for their interest, and especially Kollár for directing us to the references [Kol15] and[HW36]. The fifth author also gratefully acknowledges lively discussions with Damiano esta, where she first learned of some of the extremal properties of Frobenius forms, suchas the inseparability of their Gauss maps.2. Background on the F -pure Threshold Fix a polynomial ring k [ x , . . . , x n ] over a field of characteristic p > in any number ofvariables. Given a homogeneous form f ∈ k [ x , . . . , x n ] , we consider the singularity, at theorigin, of the hypersurface defined by vanishing of f . Algebraically stated, we considerthe singularity of f at m , where m denotes the maximal ideal h x , . . . , x n i of the origin. Definition 2.1.
The F -pure threshold of f ∈ k [ x , . . . , x n ] (at the maximal ideal m ) isthe real number fpt( f ) = sup (cid:26) Np e (cid:12)(cid:12)(cid:12)(cid:12) f N / ∈ m [ p e ] (cid:27) = inf (cid:26) Np e (cid:12)(cid:12)(cid:12)(cid:12) f N ∈ m [ p e ] (cid:27) , where m [ p e ] denotes the Frobenius power h x p e , . . . , x p e n i of m .While first explicitly defined (in a more general setting) by Takagi and Watanabe[TW04] as the "threshold c " beyond which the pair ( S, f c ) is F -pure (see [HY03]), thedefinition above is a reformulation that has evolved through the work of many authors(e.g., see [MTW05], [BMS08]). A gentle introduction can be found in the survey [BFS13].Although not obvious, the F -pure threshold is in fact a rational number [BMS08].Further basic properties, including some immediate upper and lower bounds well-knownto experts, are summarized below in the setting we will need them: Proposition 2.2.
Let f be a homogeneous form of degree d > over a field k of charac-teristic p > . Then(1) fpt( f ) ≤ .(2) For any r ≥ , we have fpt( f r ) = fpt( f ) r . (3) fpt( f ) ≥ d , with equality when f is a power of a linear form; [TW04, 4.1] .(4) If f is in n variables, then fpt( f ) ≤ nd .Proof. For (1), since f ∈ m [ p ] = m , ∈ (cid:26) Np e (cid:12)(cid:12)(cid:12)(cid:12) f N ∈ m [ p e ] (cid:27) . So the infimum of this setis at most one, always. The second statement similarly follows easily from the definition.For (3), we argue as follows. Let r e = ⌊ p e /d ⌋ − . Then f r e has degree less than p e , sois not in m [ p e ] . So ⌊ p e /d ⌋− p e ∈ (cid:26) Np e (cid:12)(cid:12)(cid:12)(cid:12) f N / ∈ m [ p e ] (cid:27) for all e , and so the supremum is at least ⌊ p e /d ⌋− p e for all e . Since these converge from below to d as e goes to infinity, the F -purethreshold is at least d .For (4), observe that m m ⊂ m [ p e ] for any m ≥ np e − n + 1 . So if f has degree d , thenfor any natural number r e such that r e ≥ np e − n +1 d , we have f r e ∈ m [ p e ] . n particular, this is true for r e = (cid:24)(cid:18) np e − n + 1 d (cid:19)(cid:25) = (cid:24) p e (cid:18) nd − ndp e + 1 dp e (cid:19)(cid:25) . This means that r e p e ∈ (cid:26) Np e (cid:12)(cid:12)(cid:12)(cid:12) f N ∈ m [ p e ] (cid:27) for all e. Since r e p e converges to nd from above, the infimum of this set is at most nd . (cid:3) The idea is that "worse" singularities have smaller F -pure threshold, just as smallerlog canonical thresholds for a complex singularity indicate "worse singularities."2.3. Connection with log canonical threshold.
The log canonical threshold is aninvariant of a complex singularity that can be defined using either integrability or Hiron-aka’s resolution of singularities. For a Q -divisor D on a smooth complex variety X , it isthe threshold beyond which the pair ( X, cD ) fails to be log canonical; see, for example,[Kol97].A form f over the complex numbers determines a collection of forms over fields ofcharacteristic p , for varying p , as follows. Let A be the finitely generated subring of C obtained by adjoining all the complex coefficients of f to Z . Interpreting the form f asan element of A [ x , . . . , x n ] , for each µ ∈ maxSpec A , we define f µ to be the image of f in the quotient ring A/µ [ x , . . . , x n ] , a polynomial ring over the finite field A/µ .The work of Hara and Yoshida [HY03] and Takagi and Watanabe [TW04] implies thatthe log canonical threshold of f is(2) sup { fpt( f µ ) | µ ∈ maxSpec A } . In particular, any lower bound on the F -pure threshold (independent of p ) implies one forthe log canonical threshold. In particular, our Theorem 3.1 implies that the log canonicalthreshold of a reduced complex form of degree d is bounded below by d − . However, thisbound is far from sharp; see paragraph 1.4.On the other hand, (2) implies that upper bounds for the log canonical threshold suggesttight upper bounds for the F -pure threshold. For example, it is easy to compute that adegree d form over C defining an isolated singularity has log canonical threshold min( nd , ,and indeed, the proof of Proposition 2.2 (4) above shows that min( nd , is what we mightexpect for the F -pure threshold of most general forms f of degree d in n variables. Thisintuition is made precise in [Her16].One research thread in the literature is concerned with understanding the extent towhich the F -pure threshold pushes up against these theoretic upper bounds. A long-standing open conjecture predicts that when f is obtained by reduction to characteristic p from a polynomial over C , then for infinitely many p , the F -pure threshold will be equalto the log canonical threshold of the corresponding complex singularity; see, for example,[MTW05, 3.6]. n this paper, we tackle the opposite question: find general lower bounds on the F -purethreshold in terms of degree and investigate the extent to which we push up against thesebounds. 3. Lower Bounds on the F -pure Threshold In this section, we establish the lower bound of our main theorem, Theorem 1.1, byproving the following:
Theorem 3.1.
Fix an arbitrary field k of characteristic p > . Let f ∈ k [ x , . . . , x n ] be ahomogeneous reduced polynomial of degree d = deg( f ) ≥ . Then (3) fpt( f ) ≥ d − Furthermore, if equality holds in (3), then d = q + 1 , where q ≥ is a power of p . Having degree p e + 1 is necessary but not sufficient to achieve the lower bound (3)on the F -pure threshold: some forms of degree p e + 1 are more singular than others.Characterizing these "maximally singular" forms where the lower bound is achieved isthe task of the next section; see Theorem 4.3. Remark 3.2.
For the bound (3), a careful reading of the proof shows that our hypothesiscan be weakened from "reduced" to "not a power of a linear form." But for powers oflinear forms, of course, the lower bound is false: the F -pure threshold of x d is d .For the subsequent statement about what happens when the lower bound is achieved,however, we need the "reduced" hypothesis. For example, x y has F -pure threshold = d − in every characteristic, but is not a power of p . Remark 3.3.
The main theorem of [BS15] can be interpreted to give a lower bound of − d − p on the F -pure threshold in the very special case where the degree d of the form f is equal to the number of variables n , the projective hypersurface defined by f is smooth,and the characteristic p > n = d . (See also [Mül18] for the quasi-homogeneous case.)However, because of the restriction that p > d , the Bhatt-Singh bound never applies inour extremal case.Before beginning the proof of Theorem 3.1, we need a few lemmas. The first allows usto assume the ground field is algebraically closed. Lemma 3.4.
Let k ⊂ k ′ be any field extension, of characteristic p > . For any f ∈ k [ x , . . . , x n ] , the F -pure threshold is independent of whether we view f as a polynomialover k or over k ′ .Proof. Let A = k [ x , . . . , x n ] have homogeneous maximal ideal m A and B = k ′ [ x , . . . , x n ] have homogeneous maximal ideal m B . Note that A ⊂ B is faithfully flat so that IB ∩ A = I for any ideal I of A . ow, if f N ∈ m [ p E ] A , then the same is true in B . So the F -pure threshold over A is at leastthe F -pure threshold over B . But conversely, if f N ∈ m [ p E ] B = m [ p E ] A B , then intersectingwith A , we have f N ∈ m [ p E ] A B ∩ A = m [ p E ] A . This gives the reverse inequality. (cid:3) The next is a codification of a well-known trick we use many times, whose statementwe make explicit for the convenience of the reader. The proof we leave as an exercise.
Lemma 3.5.
Let y , . . . , y n be a regular sequence in a commutative ring. Suppose thereexists an element g , and natural numbers a i ≤ N i such that ( y a y a . . . y a n n ) g ∈ h y N , y N , . . . , y N n n i . Then g ∈ h y N − a , y N − a , . . . , y a N − a n n i . The next two lemmas will be used to reduce Theorem 3.1 to the case of two variables.
Lemma 3.6.
Let f be a homogeneous polynomial in k [ x , . . . , x n ] . For any linear form L not dividing f , let f denote the image of f in k [ x , . . . , x n ] / h L i ∼ = k [ x , . . . , x n − ] . Then fpt( f ) ≥ fpt( f ) . Proof.
Suppose f N ∈ m [ p e ] . Then also f N ∈ m [ p e ] . So we have an inclusion of sets (cid:26) Np e (cid:12)(cid:12)(cid:12)(cid:12) f N ∈ m [ p e ] (cid:27) ⊆ (cid:26) Np e (cid:12)(cid:12)(cid:12)(cid:12) f N ∈ m [ p e ] (cid:27) . So the infimum of the left-hand set is at least the infimum of the right-hand set. That is, fpt( f ) ≥ fpt( f ) . (cid:3) Remark 3.7.
Lemma 3.6 can be used to give a quick proof of the easy lower bound fpt( f ) ≥ d , where d = deg( f ) , shown Proposition 2.2. Namely, by modding out n − linearly independent one-forms, we eventually reduce to the one-variable case, and have fpt( f ) ≥ fpt( x d ) = d .Finally, we need the following restatement of a Bertini-type theorem for reduced vari-eties: Lemma 3.8.
Consider a polynomial ring k [ x , . . . , x n ] in at least three variables over aninfinite field. Fix a reduced form f in k [ x , . . . , x n ] , and for any linear form L , let f denotethe image of f in the quotient ring k [ x , . . . , x n ] / h L i ∼ = k [ x , . . . , x n − ] . If L is sufficientlygeneral, then the polynomial f is also reduced in the polynomial ring k [ x , . . . , x n − ] .Proof. This is essentially a restatement of [FOV99, 3.4.14], which implies that (over aninfinite field) a general hyperplane section of a reduced scheme X ⊂ P n − is reduced:because f is reduced, the projective variety defined by f in P n − is reduced, so alsothe general hyperplane section—namely, the variety defined by f and L —is reduced.Because the homogeneous coordinate ring of this hyperplane section is isomorphic to k [ x , . . . , x n ] / h L, f i ∼ = k [ x , . . . , x n − ] / h f i , we see that also f is reduced. (cid:3) roof of Theorem 3.1. By Lemma 3.4, we may assume the ground field is algebraicallyclosed.We first reduce to the case of two variables. Given a form f of degree d in n variables,we can successively mod out a sequence of n − of independent linear forms until wehave a form f of degree d in two variables. Now Lemma 3.6 implies that any lower boundon the F -pure threshold of f is also a lower bound for f . Likewise, if this lower boundis achieved for f , then it is also achieved for f . So because d and p are the same for f and f , it suffices to prove the required implication for f . Finally, if f is reduced, Lemma3.8 ensures that f is reduced, by choosing a sufficiently general sequence of linear forms.Thus the proof of Theorem 3.1 reduces to the case of two variables.We next dispose of the case d = 2 . Any reduced polynomial of degree 2 in two variablesfactors into two distinct linear factors, so without loss of generality, f = xy . By directcomputation, the F -pure threshold is in every characteristic. So equality holds in (3)and d = p + 1 .Now assume d ≥ . Given a form f of degree d in two variables, we can factor as f = xyg where g is a form of degree d − . To prove the lower bound (3), it suffices toshow that for any p E such that f N ∈ m [ p E ] , we have(4) Np E ≥ d − . But if f N ∈ m [ p E ] = h x p E , y p E i , then writing ( xyg ) N = Ax p E + By p E for some homogeneous A and B , we can use the fact that { x, y } is a regular sequence(Lemma 3.5) to see that g N ∈ h x p E − N , y p E − N i . By comparing degrees, it follows that N ( d − ≥ p E − N , which is equivalent to thedesired inequality (4). This shows the F -pure threshold is at least d − .We now investigate what happens when equality holds in (3). Assume that fpt( f ) =1 d − . For all e ≥ , we have ⌈ p e +1 d − ⌉ p e ≥ p e + 1 p e ( d − > d − f ) . Since fpt( f ) is the supremum of the set { Np E | f N m [ p E ] } , and ⌈ pe +1 d − ⌉ p e is strictly biggerthan this supremum, it must be that f ⌈ pe +1 d − ⌉ ∈ m [ p e ] . To ease notation, we set K e = ⌈ p e +1 d − ⌉ . We have f K e ∈ h x p e , y p e i , so we write(5) f K e = Ax p e + By p e where A and B are forms of degree dK e − p e . lso using the strict inequality K e p e > d − , we have K e p e − p E > d − for all sufficientlylarge E . So similarly, f p E − e K e − ∈ h x p E , y p E i , and we can write(6) f p E − e K e − = Cx p E + Dy p E for some forms C and D .Now raising (5) to the power p E − e and multiplying (6) by f , we have two differentexpressions for the form f p E − e K e . Subtracting them, we have(7) ( A p E − e − f C ) x p E + ( B p E − e − f D ) y p E = 0 . Again using the fact that x, y is a regular sequence, we conclude that(8) ( A p E − e − f C ) ∈ h y p E i . But we claim that(9) deg( A p E − e − f C ) = p E − e deg A < p E , which implies that A p E − e − f C = 0 .To check claim (9), recall that the degree of A is dK e − p e , so that (9) is equivalent to dK e < p e . In turn, we have dK e = d (cid:24) p e + 1 d − (cid:25) ≤ d (cid:18) p e + 1 d − (cid:19) , which is less than p e for all large e .Having established the veracity of claim (9) for E ≫ e ≫ , we can conclude using theinclusion in (8) that A p E − e − f C = 0 , so that A p E − e = f C. We now invoke the fact that f is a product of distinct irreducible polynomials: the uniquefactorization property of the polynomial ring implies that f must divide the form A .Similarly, f divides B .Now, because f divides both A and B , we can divide f out of both sides of equation(5) above, to get(10) f K e − ∈ h x p e , y p e i . Remembering that f = xyg , where g has degree d − , we can again use the fact that x, y is a regular sequence (Lemma 3.5) to deduce that(11) g K e − ∈ h x p e − K e +1 , y p e − K e +1 i . Looking at degrees, this says that ( K e − d − ≥ p e − K e + 1 , which is equivalent to ( d − K e ≥ p e + d − , r equivalently,(12) K e ≥ p e + d − d − p e + 1) + ( d − d − p e + 1 d − d − d − . Remembering that K e = ⌈ p e +1 d − ⌉ , we see that inequality (12) can hold only if p e + 1 d − is not an integer and p e + 1 d − p e + 1 d − d − d − round up to the same integer. This means that p e + 1 d − must be equal to (cid:22) p e + 1 d − (cid:23) + 1 d − ;put differently, the remainder when we divide p e + 1 by d − is 1. So d − divides p e . Inthis case, d − is a power of p (as desired). The proof is complete. (cid:3) Extremal Singularities
In the previous section, we proved following general bound on the F -pure threshold ofany reduced form f of degree d : fpt( f ) ≥ d − . In this section, we complete the proof of Theorem 1.1, by characterizing of those polyno-mials for which the minimal possible F -pure threshold is achieved. Definition 4.1.
A reduced form f of degree d > is called an extremal singularity ifits F -pure threshold is equal to d − . Remark 4.2.
The case where d = 2 is a somewhat special case that we have chosento exclude from Definition 4.1. Thinking of the F -pure threshold as a measure of howfar a singularity is from being F -pure, an extremal singularity is one that is as far astheoretically possible, according the lower bound above, from being F -pure. The F -purethreshold of a reduced form of degree two is 1 in every characteristic, so while forms ofdegree two achieve the theoretical lower bound, they also achieve the theoretical upperbound on the F -pure threshold (Proposition 2.2). Thus they are both the "most singular"and the "least singular" type of singularity we can have in degree two! Among (reduced)multiplicity two singularities, no finer gradation of singularities exists. Theorem 4.3.
Let f be a reduced form of degree d over a field of positive characteristic p . Then the F -pure threshold of f is d − if and only if f can be written (13) x p e L + x p e L + · · · + x p e n L n for some e ≥ , where the L i are linear forms. Forms of the type (13) have many extremal properties besides the one guaranteed byTheorem 4.3, so they warrant a name:
Definition 4.4. A Frobenius form is any form of degree q + 1 in the ideal m [ q ] = h x q , . . . , x qn i , where q is a power of the characteristic. he proof of Theorem 4.3 relies on the following lemma justifying the intuition thatpolynomials in "Frobenius powers" m [ p e ] = h x p e , . . . , x p e n i are "more singular" than poly-nomials not in m [ p e ] : Lemma 4.5.
The F -pure threshold of f is less than or equal to p e if and only if f ∈ m [ p e ] .In fact, if f m [ p e ] , then fpt( f ) ≥ p e + p e .Proof. First assume f ∈ m [ p e ] . Then p e is in the set (cid:26) Np E (cid:12)(cid:12)(cid:12)(cid:12) f N ∈ m [ p E ] (cid:27) . So the F -purethreshold, which is the infimum of this set, is bounded above by p e .For the converse statement, first observe that S = k [ x , . . . , x n ] can be viewed as afree module over the subring S p e ′ for all e ′ . Indeed, we can take { λx m } as a basis, where λ ranges over a basis for k over k p e ′ and x m ranges through all monomials in the x i .Moreover, if f / ∈ m [ p e ′ ] , then it can be taken to be a part of a free basis for S over S p e ′ —for example, taking any monomial x m ′ that appears in f with all exponents less than p e ′ ,we can replace x m ′ in the basis { λx m } by f to get another basis. In particular, projectiononto the S p e ′ -submodule generated by f gives us an S p e ′ -linear map π : S → S p e ′ sending f to 1.Now, assume f / ∈ m [ p e ] . Then for all e ′ , the flatness of Frobenius [Kun69] implies thatalso f p e ′ / ∈ m [ p e + e ′ ] . Furthermore, for all e ′ ≥ e , we have f p e ′ +1 = f · f p e ′ / ∈ m [ p e + e ′ ] , for otherwise, we could apply π from the previous paragraph to see that f p e ′ ∈ m [ p e + e ′ ] .This means that the rational number p e ′ +1 p e + e ′ is in the set (cid:26) Np e (cid:12)(cid:12)(cid:12)(cid:12) f N / ∈ m [ p e ] (cid:27) for all e ′ ≥ e .So the supremum of this set is at least p e + p e + e ′ for all e ′ ≥ e . The largest of these iswhen e = e ′ , so the supremum is at least p e + p e . This completes the proof. (cid:3) Proof of Theorem 4.3.
Both directions follow from facts we have already established.First, we claim that a Frobenius form f = x p e L + x p e L + · · · + x p e n L n has F -purethreshold d − = p e . Indeed, fpt( f ) ≥ p e by Theorem 3.1, whereas fpt( f ) ≤ p e by Lemma4.5.For the other direction, suppose that fpt( f ) = d − for some reduced form f . NowTheorem 3.1 tells us that d = p e + 1 for some e , which means that fpt( f ) = d − = p e forsome e . Now Lemma 4.5 guarantees that f ∈ m [ p e ] . Thinking about degrees, we see that f must be of the form (13). (cid:3) . Matrix Representation of Frobenius Forms
Our next goal is to study Frobenius forms, the polynomials defining extremal singular-ities. As we proved in the previous section, these are forms(14) X x p e i L i , where the L i are linear forms. A Frobenius form can be uniquely factored as(15) h = (cid:2) x p e x p e . . . x p e n (cid:3) A x x ... x n , where A is the n × n matrix whose i -th row is made up of the coefficients of the linearform L i . This allows us to use linear algebra to conveniently study Frobenius forms.Let’s look at how changing coordinates affects the matrix representing a Frobeniusform. For a matrix B of any size, we denote by B [ p e ] the matrix obtained by raising allentries to the p e -th power. If g is a change of coordinates represented by an invertible n × n matrix, then g · x p e x p e ... x p e n = g [ p e ] x p e x p e ... x p e n = [ g x x ... x n ] [ p e ] . Here the notation · indicates the ring automorphism induced by the linear change ofcoordinates, and all other adjacent symbols are usual matrix product.So our change of coordinates formula for g acting on h is g · (cid:2) x p e x p e . . . x p e n (cid:3) A x x ... x n = (cid:2) x p e x p e . . . x p e n (cid:3) (cid:2) g [ p e ] (cid:3) tr Ag x x ... x n , where the superscript " tr " indicates the transpose. We can write this in the compactform g · [( ~x [ p e ] ) tr A ~x ] = ( ~x [ p e ] ) tr (cid:2) g [ p e ] (cid:3) tr Ag ~x.
That is, if h is a Frobenius form represented by the matrix A , then the Frobenius form g · h , where g is any linear change of coordinates, is represented by the matrix (cid:2) g [ p e ] (cid:3) tr Ag .It is worth recording, for future reference, how each elementary coordinate operationaffects the matrix A representing a Frobenius form. assuming e > emma 5.1. Let (16) h = (cid:2) x p e x p e . . . x p e n (cid:3) A x x ... x n , be a Frobenius form. Then elementary linear changes of coordinates are reflected in A asfollows: • Swapping two variables ( x i ↔ x j ), fixing the others changes A by swapping columns C i and C j and rows R i and R j , fixing the others. • Multiplying coordinate x i by a non-zero scalar λ ( x i λx i ), fixing the otherschanges A by multiplying row R i by λ p e and column C i by λ . • Replacing x i by x i + λx j for some j = i , fixing the others changes A by replacingcolumn C j by column C j + λC i and row R j by row R j + λ p e R i . Embedding dimension, rank and the singular locus.
A form f ∈ k [ x , . . . , x n ] is non-degenerate if it can’t be written as a polynomial in fewer variables after anylinear change of coordinates. In this case, we say that the singularity defined by f has embedding dimension n . [In general, the embedding dimension of a singularity is thedimension of the Zariski cotangent space m/m of the maximal ideal of the local ring atthe singularity.]Another useful invariant of a Frobenius form is the rank , by which we mean the rankof the representing matrix. The following proposition implies that the rank of a Frobeniusform is the same as the codimension of the singular locus of the corresponding extremalsingularity. Proposition 5.3.
The singular locus of an extremal singularity defined by the Frobeniusform (cid:2) x p e x p e · · · x p e n (cid:3) A x x ... x n is the p e -fold linear subvariety defined by the equations A tr x p e x p e ... x p e n = 0 . Put differentially, the (reduced) singular set is the linear space defined as the kernel ofthe matrix [ A [1 /p e ] ] tr , where ( A [1 /p e ] ) tr is the transpose of the matrix whose entries are the p e -th roots of the entries of A . roof. Write h = x p e L + x p e L + · · · + x p e n L n where the coefficients of the linear forms L i are given by the rows of A = [ a ij ] . Thesingular locus is defined by the vanishing of the partial derivatives ∂h∂x j . But for each j , ∂h∂x j = x p e a j + · · · + x p e n a nj = (cid:2) a j · · · a nj (cid:3) x p e ... x p e n = h a pe j · · · a pe nj i x ... x n p e , so the proposition follows. (cid:3) The following consequence is immediate:
Corollary 5.4.
The Hessian of a Frobenius form is zero. That is, ∂ h∂x i ∂x j = 0 for all ≤ i ≤ j ≤ n . Finally, we record a simple lemma which gives a nice form for a Frobenius form in termsof its rank and embedding dimension.
Lemma 5.5.
A Frobenius form of rank r can be written, in suitable coordinates, as h = x p e L + x p e L + · · · + x p e r L r , where the L i are linearly independent linear forms. In this case, the embedding dimension n is equal to the dimension of the space spanned by the forms x , x , . . . , x r , L , . . . , L r .In particular, r ≥ n .Proof. Let A denote the matrix representing h . Swapping variables, assume the first r rows are linearly independent. Because the rows beyond the r -th are all dependent on thefirst r , a suitable sequence of row operations can be used to transform these bottom rowsinto zero rows; the corresponding column operations (Lemma 5.1) do not affect these zerorows. Thus without loss of generality, we can assume the bottom n − r rows of A arezero rows. This implies that h can be written as x p e L + x p e L + · · · + x p e r L r for somelinear forms L i . The L i are independent because they span the row space of A . For theembedding dimension, note that if x , x , . . . , x r , L , . . . , L r span a space of dimension lessthan n , then h can be written in fewer than n variables, so it is degenerate (that is, theembedding dimension is not n ). (cid:3) Remark 5.6.
A very special kind of Frobenius form is a
Hermitian form of characteristic p . These are Frobenius forms in which the matrix A representing the form satisfies a ij = a pji for all i, j (in particular, since this implies a q ij = a ij for all i, j , a Hermitian formis defined over the finite field F q and the Frobenius map ( q -th power map) is an involutionthat plays a role analogous to complex conjugation). Hermitian hypersurfaces—projectivehypersurfaces defined by Hermitian forms—have well-studied "extremal" properties, suchas an abundance of rational points; see [BC66], [Seg65], and [HK16].Hermitian forms are classically known to be "diagonalizable;" see [HW36] or [BC66,4.1]. In the full rank case, every Frobenius form is projectively equivalent (over k ) to a ermitian form, by our Theorem 6.1, which shows full rank Frobenius forms are diago-nalizable. In lower rank, Hermitian forms constitute a strictly proper class of Frobeniusforms; see Remark 7.7.6. Extremal Singularities of Full Rank
In this section, we prove the following characterization of isolated extremal singularities.
Theorem 6.1.
Every full rank extremal singularity over an algebraically closed field k ofcharacteristic p > is represented, in suitable linear coordinates, by the diagonal form x q +11 + · · · + x q +1 n , where q is some power of p . We prove Theorem 6.1 using only basic polynomial algebra and Hilbert’s Nullstellen-satz. In the special case of a
Hermitian form —that is, where the matrix representingthe Frobenius form satisfies a ij = a pji for all i, j —Theorem 6.1 can be found in [HW36];see also [BC66] for a modern approach. In the special case where the Frobenius formhas embedding dimension at least four, Theorem 6.1 follows from the main theorem of[Bea90]. Kollár suggested an alternate proof as well; see Remark 6.3.
Proof.
We will prove this by induction on n . The case where n = 1 is trivial.Let h be a full rank Frobenius form in n variables for any n ≥ . Write h as h = (cid:2) x p e x p e · · · x p e n (cid:3) A x x ... x n where A is an n × n matrix over k . The induction happens by showing that we can changecoordinates to put A into the block form(17) ∗ ∗ · · · ∗ ∗ ∗ · · · ∗ ... ... . . . ... ... ∗ ∗ · · · ∗
00 0 · · ·
Equivalently, this says we can write the Frobenius form as h = x p e +1 n + g ( x , . . . , x n − ) , where g is full rank Frobenius form in the first n − variables. So if we know by inductionthat g can be put into the desired form by a linear change of coordinates involving onlythe variables x , . . . , x n − , then it follows that h is in this form as well.We will use the following lemma: There appears to be some confusion in the literature interpreting the 1936 paper of Hasse and Witt(see, e.g., the "warning and request" in [AH19]). emma 6.2. If a full rank Frobenius form (18) h = (cid:2) x p e x p e · · · x p e n − x p e n (cid:3) ∗ ∗ · · · ∗ a n ∗ ∗ · · · ∗ a n ... ... . . . ... ... ∗ ∗ · · · ∗ a n − ,n a n a n · · · a n,n − a nn x x ... x n − x n satisfies (19) a in a p e − nn = a p e ni for all i = 1 , . . . , n − , then we can change coordinates to put h in the block form (17). That is, we can changecoordinates to get h in the form x p e +1 n + g where g is a Frobenius form in x , x , . . . x n − .Proof. Note that if a nn = 0 is zero, then the condition (19) implies that the last row iszero, contrary to the full rank assumption on A . So a nn = 0 . We can therefore assume,without loss of generality, that a nn = 1 . Indeed, scaling x n by a ( p e + 1) -th root of a − nn (call it c ) changes the matrix A by multiplying row n by c p e and column n by c (seeLemma 5.1). This allows us to assume that a nn = 1 without destroying condition (19).Now, assuming a nn = 1 , the change of coordinates that sends x n x n − a n x − a n x − · · · − a n,n − x n − and fixes x i for ≤ i ≤ n − gives us the desired form. One simply checks that substituting x n − a n x − a n x − · · · − a n,n − x n − for x n into h produces a polynomial of the form x p e +1 n + g ( x , . . . , x n − ) . Alternatively, invoking Lemma 5.1, because of the special formof A , we see that subtracting a ni times column n from column i will place a zero in the i -th column of the final row, while the corresponding row operation also makes the i -throw of the final column zero. (cid:3) Continuing the proof of Theorem 6.1 now armed with Lemma 6.2, we note that itsuffices to show that any full rank Frobenius form can be put in the form (18). Let h = (cid:2) ~x [ p e ] (cid:3) tr A (cid:2) ~x (cid:3) be an arbitrary Frobenius form. Suppose g is a change of coordinatematrix with indeterminate entries. Changing coordinates, the new matrix of g · h is(20) ˜ A = ( g [ p e ] ) tr Ag.
We need to show that there is a choice of g such that the entries of ˜ A satisfy the hypothesisof Lemma 6.2.Thinking of the entries of g as indeterminates Y ij , the matrix product (20) has entries(21) ˜ A ij = X ≤ k,ℓ ≤ n a ℓk Y p e ℓi Y kj , which are homogeneous polynomials in the Y ij . It suffices to prove that there exist valuesof the Y ij that satisfy the equations(22) ˜ A in ˜ A p e − nn = ˜ A p e ni for all i = 1 , , . . . , n − , nd for which the matrix g has a non-zero determinant.Plugging in the expressions (21), the equations (22) become(23) ˜ A p e − nn X ≤ k,ℓ ≤ n a ℓk Y p e ℓi Y kn ! = X ≤ k,ℓ ≤ n a ℓk Y p e ℓn Y ki ! p e i = 1 , , . . . , n − , each of which can be rearranged into a linear equation in Y p e i , Y p e i , . . . , Y p e ni :(24) F Y p e i + F Y p e i + · · · + F n Y p e ni = 0 i = 1 , . . . n − , where the coefficients F j of Y p e ji are F j = ˜ A p e − nn n X k =1 a jk Y kn − n X ℓ =1 a p e ℓj Y p e ℓn ! j = 1 , . . . , n. The key things to notice here are that the coefficient F j of Y p e ji in the equations (24) isthe same for each i = 1 , . . . , n , and that it is a polynomial in only Y n , . . . , Y nn , the entriesof the last column of the matrix g . Thus the F , . . . , F n form a homogenous system ofpolynomials in the n indeterminates Y n , . . . , Y nn of degree p e .We claim that there is a choice of values for Y n , . . . , Y nn , not all zero, for which all F , . . . , F n vanish. In this case, we can take g to be the matrix that has these values asits final column, with any choice of the first n − columns that makes g invertible. Forthis choice of g , we will have proven that changing coordinates by g , the form h can beput into the desired form of Lemma 6.2. Thus the proof is complete once we have founda non-zero solution to the system { F = F = · · · = F n = 0 } .To prove this claim, we invoke Hilbert’s Nullstellensatz: provided the ideal generated by F , F , . . . , F n in k [ Y n , Y , . . . , Y nn ] is not h Y n , Y , . . . , Y nn i -primary, the Nullstellensatzprovides the needed non-zero solution. But expanding out the vacuously true expression ˜ A nn ˜ A p e − nn = ˜ A p e nn produces the following relation: Y p e n F + Y p e n F + · · · + Y p e nn F n = 0 . Since Y p e nn has degree p e , it cannot be in the ideal generated by the elements F , F , . . . , F n − ,which have degree p e , showing that { F , F , . . . , F n } is not a regular sequence. Thus theideal h F , F , . . . , F n i has height strictly less than n . Thus the Nullstellensatz gives theneeded non-zero solution to the system F = F = · · · = F n = 0 . The theorem isproved. (cid:3) Remark 6.3.
János Kollár suggested a different argument for Theorem 6.1 based onshowing the stabilizer of the GL n -action on the space of Frobenius forms is zero dimen-sional. 7. Isomorphism Types of Extremal Singularities.
We say that two extremal singularities are isomorphic if they differ by a linear changeof coordinates. n the previous section, we proved that there is at most one extremal singularity with isolated singularity , up to isomorphism, in each dimension and degree over k . Allowing alarger singular locus, now, we have the following. Theorem 7.1.
Over a fixed algebraically closed field, there are finitely many isomorphismtypes of extremal singularities of bounded degree and dimension.
Theorem 7.1, as well as the explicit bound in Corollary 7.4, are both immediate corol-laries of the following more precise theorem ensuring that every Frobenius form has arepresenting matrix of an especially nice "sparse" form:
Theorem 7.2.
Fix any algebraically closed field of characteristic p > . A Frobeniusform (of, say, degree p e + 1 ) of embedding dimension n and rank r can be represented bya matrix with the following properties:(1) All rows beyond the r -th are zero.(2) All columns beyond the n -th are zero.(3) There are exactly r non-zero entries (all of which are 1) occurring in positions (1 j ) , (2 j ) , . . . , ( r j r ) , where j > j > · · · > j r . In particular, Theorem 7.2 says that a Frobenius form of rank r is represented by amatrix whose columns are unit column vectors e r , e r − , . . . , e (in that order), interspersedwith zero columns. There are only finitely many such matrices of fixed size. Hence thereare only finitely many Frobenius forms in N variables up to change of coordinates, showingTheorem 7.1 follows from Theorem 7.2. Definition 7.3.
A Frobenius form, or its matrix, is said to be in sparse form if itsmatrix satisfies conditions (1), (2), and (3) of Theorem 7.2.Before proving Theorem 7.2, we note that it provides the following explicit bound:
Corollary 7.4.
Fix an algebraically closed field k of positive characteristic. The numberof isomorphism types of extremal singularities of fixed embedding dimension n and fixed(arbitrary) degree is bounded above by the n -th Fibonacci number F n .Proof of Corollary. Let h be a Frobenius form of rank r and embedding dimension n . ByTheorem 7.2, h can be represented by a matrix A in sparse form. This means that h = x p e L + . . . + x p e r L r , where L , . . . L r ∈ { x , . . . , x n } are variables that appear in reverse order, each variableappearing at most once.All the variables x r +1 , . . . , x n must appear in the list L , . . . , L r (otherwise the embed-ding dimension is less than n ). Therefore, we can assume that L = x n , L = x n − , · · · , L n − r = x r +1 , By definition, the Fibonacci sequence is defined inductively by F = F = 1 and F n +1 = F n + F n − . n light of property (3) of sparseness. For the remaining linear forms { L n − r +1 , . . . , L r } ,we can choose r − n variables out of the remaining variables { x , . . . , x r } . There are (cid:18) r r − n (cid:19) = (cid:18) rn − r (cid:19) such choices.So if N ( n, r ) denotes the number of isomorphism types of Frobenius forms (of fixeddegree over a fixed field) of embedding dimension n and rank r , we have N ( n, r ) ≤ (cid:0) rn − r (cid:1) .Summing over all ranks r , the total number of Frobenius forms of fixed degree over k isbounded above by n X r =1 N ( n, r ) ≤ n X r =1 (cid:18) rn − r (cid:19) = F n where the last equality is well-known; see, for example, Lucas’s 1891 text [Luc91, p7]. (cid:3) Remark 7.5.
The bound of Corollary 7.4 agrees with the actual number of non-degenerateFrobenius forms in small numbers of variables; for example, there is one of embedding di-mension one, two of embedding dimension two, three of embedding dimension three, andfive of embedding dimension four. This was proved, essentially, in [KKP + n = 5 or more, the bound is strict. See [GKK + ] for details. Proof of Theorem 7.2.
Fix a Frobenius form in N variables (of some degree d = p e + 1 ).We induce on N .For N = 1 , there is one: x p e +1 . The corresponding matrix [1] satisfies the neededconditions.For N = 2 , there are two more, both of embedding dimension two—one of rank tworepresented by x p y + y p x (Theorem 6.1) and one of rank one represented by x p e y (Lemma5.5). Remembering also the degenerate form x p e +1 of embedding dimension one, thecorresponding matrices are (cid:20) (cid:21) , (cid:20) (cid:21) , and (cid:20) (cid:21) , all of which satisfy the three conditions of Theorem 7.2.Now assume the Frobenius form h is in N ≥ variables. Note that if N exceeds theembedding dimension n , then we can eliminate a variable from h to get the form h ′ in N − variables of the same rank and embedding dimension. By induction, h ′ is represented bya matrix A ′ in the sparse form. Enlarging A ′ by appending an additional row and columnof zeros produces a matrix for h that has the required properties. So, we might as wellassume that A is non-degenerate—that is, we henceforth assume that N = n .Say the rank of h is r . If r = n = N , then the form is full rank and is representedby any full rank n × n matrix by Theorem 6.1. In particular, the "reverse permutationmatrix" A = (cid:2) e n e n − · · · e e (cid:3) represents h and is in sparse form. So we assumethe rank r is strictly less than n . The first few terms are usually zero using the usual convention that (cid:0) ab (cid:1) = 0 when b > a . This isconsistent with Lemma 5.5: the rank is at least half the embedding dimension always. he form h can be written x p e L + · · · + x p e r L r , where the L i are linear forms, byLemma 5.5. This says that the corresponding matrix has zero rows beyond the rank, andin particular, since r < n , the bottom row of A is zero. We can also assume a n = 1 , since h is non-degenerate.Now adding multiples of row 1 to the other rows, we can assume the (rest of the) entriesin the last column of A are all zero. The corresponding column operations (as prescribedby Lemma 5.1) add multiples of column 1 to columns , , . . . , n − ; this may changethe entries of A in the first n − columns, but not in the last column. Then we can addmultiples of column n to the other columns to clear out the (rest of the) first row of A ;the corresponding row operations add multiples of row n to other rows, which changesnothing since row n is a zero row. That is, without loss of generality, we can assume that A has the block form(25) . . . ∗ a a . . . a ,n − ∗ a a . . . a ,n − ... ... ... . . . ... ... ∗ a n a n . . . a n,n −
00 0 0 . . . = E B where B is an ( n − × ( n − matrix (consisting of the "middle" n − rows and columnsof A ), each block is an either a row or column of length n − , and E is a column matrixof length n − .The "middle" matrix B represents a Frobenius form in n − variables. So we can applythe inductive hypothesis to assume that B has the sparse form of Theorem 7.2. This isachieved by a change of coordinates involving only the variables { x , . . . , x n − } , amountingto row and column operations on A involving only rows and columns , . . . , n − . Inparticular, the zeros in the matrix (25) are left unchanged (though the entries of E maybe different). Note that because A has rank r , the sparse matrix B must have rank either r − or r − . We treat the two cases separately.Case 1: Rank B is r − . In this case, the first column of A is dependent on the remainingcolumns. Thus, we can add multiples of columns through n − to column 1 to transformthe column matrix E into the zero matrix. The corresponding row operations change onlythe first row of A (see Lemma 5.1). To repair the damage done to the first row of A ,we add multiples of the last column to the other columns to successively transform eachelement in the the first row of A (except in the last spot) into zero; the corresponding rowoperations only add the zero row to the others. Note that all of these operations leave B unchanged. After this, the matrix A has the form B , where each represents a -block of the appropriate dimensions. This satisfies the needed conditions for Theorem7.2.Case 2: Rank B is r − . In this case, columns 2 through n − of A are standard columnvectors e , . . . , e r − (and zero columns). Again, we can add multiples of these columns to he first column to clear out all non-zero entries of up to and including the r − -st. Thecorresponding row operations introduce non-zero elements to the first row, but as before,these can be cleared out by adding column n to the previous columns. None of theseoperations change any entry of B . There may still be non-zero entries in column 1, butonly in rows r, r + 1 , . . . , n − . So A can be assumed of the form (25) with B rank r − in sparse form, and the first column having non-zero entries only in rows r and higher.It now suffices to show that we can change coordinates to make the first column of A the unit column e r while keeping B sparse. We will do this in two steps: Step 1:
Change the first column of A so it has only one non-zero entry, without changingany other entry of A . Step 2:
Change the first column of A so that it becomes e r , possibly replacing B withanother matrix of the same rank in sparse form.Both steps require the following lemma, whose proof we leave until the end: Lemma 7.6.
Suppose A is an n × n matrix of the form E B , where B is a rank r − matrix in sparse form, denotes either a zero row or zero columnof length n − , and E is a column vector of length n − such that the first column of A has zero entries in all rows i with i < r . Then for each row index i in the range r ≤ i ≤ n − , there is a unique sequence of row indices i , i , . . . , i m satisfying(1) Entry ( i ℓ i ℓ − ) of A is non-zero for all ℓ > .(2) Each i ℓ ∈ { , . . . , r − } for all ℓ > .(3) The indices i , i , . . . , i m are distinct.(4) Column i m is .Furthermore, if i = j , then the sets { i , i , . . . , i m } and { j , j , . . . , j m ′ } are disjoint. Details for Step 1:
We will show that a carefully-chosen sequence of elementarychanges of coordinates of the type x k x k + λx ℓ (fixing the other variables) will reducethe number of non-zero entries in column 1 by one without changing any other entry of A . By induction, therefore, Step 1 will be complete.Suppose that i and j are the row indices of two non-zero entries of column 1. Ourstrategy is to add a multiple of row i to row j to clear the non-zero entry in position ( j . The corresponding column operation (according to Lemma 5.1) adds a multiple ofcolumn i to column j . So if column i is zero , this process does not change any entryof A other than the ( j -entry, which is now zero, and we have reduced the number ofnon-zero entries of column 1 by one, completing Step 1. owever, if column i is not zero, then the corresponding column operation does changethe matrix A elsewhere. Our hypothesis on B , however, ensures that this change is limited:exactly one entry changes from zero to non-zero. Explicitly, if the unique non-zero entryof column i is in position ( i i ) , then the new unwanted non-zero entry is in position ( i j ) . Like the carnival game "whack-a-mole," we knocked out one non-zero entry (inposition ( j ) using a carefully chosen row operation only to see another non-zero entry(in position ( i j ) ) appear because of the corresponding column operation. Abbreviatingthis pair of row/column operations by " i to j ", we summarize this process with thenotation ( j i to j −−−−→ ( i j ) , highlighting the positions of the unwanted non-zero entries. Note that the pair of row/columnoperations " i to j " is the matrix description (see Lemma 5.1) for the change of coordi-nates replacing x i by x i − λ /q x j where λ = a j a i (fixing all other variables).Our strategy is to repeat this process, eliminating each new non-zero entry with somenew row operation (whose corresponding column operation may introduce one new non-zero entry) until eventually we perform a row operation whose corresponding columnoperation is simply adding a zero column. The needed sequence of row operations ischosen carefully with the help of Lemma 7.6. To this end, we let i , . . . , i m and j , . . . , j m ′ be the two disjoint sequences of row indices guaranteed by Lemma 7.6. Without loss ofgenerality, assume that m ≤ m ′ , so that column i m is zero, but columns i k , j k are not zerofor k < m . The needed sequence of elementary operations is slightly different, dependingon the parity of m : The case where m is odd: In this case, begin by adding a multiple of row i to row j to clear position ( j . The corresponding column operation adds a multiple of column i to column j . If column i is zero, therefore, we are done: we have eliminated one of thenon-zero entries of the first column without changing any other element of A . Otherwise,column i is not zero, so the corresponding column operation introduces an unwantednon-zero entry in position ( i j ) .Now we attempt to clear position ( i j ) by adding a multiple of row j (note thatcolumn j has a non -zero entry in row j , since m ′ ≥ m ). This row operation zeros outposition ( i j ) , but the corresponding column operation introduces a new non-zero entryin position ( j i ) . To correct the unwanted ( j i ) entry, we add a multiple of row i (positon ( i i ) is non-zero since m is odd). We continue this process of adding multiplesof suitable rows to clear out non-zero entries, alternating between adding i -indexed rowsand j -indexed rows (and their corresponding columns). This process produces a sequenceof mutations of A in which a single unwanted non-zero entry beginning in position ( j moves through the following positions without altering any other entry of A : ( j i to j −−−−→ ( i j ) j to i −−−−→ ( j i ) i to j −−−−→ · · · j m − to i m − −−−−−−−−→ ( j m i m − ) . Because m is odd, the final row operation in this sequence adds a multiple of row i m torow j m to clear the non-zero entry ( j m i m − ) . The corresponding column i m is zero (byassumption; see Lemma 7.6), so no new non-zero entry is created by the corresponding olumn operation. The resulting matrix has one fewer non-zero entry in column 1 but isotherwise unchanged from A . The case where m is even is similar, but we must begin by adding a multiple of row j to zero out position ( i . This ensures that our last step will involve adding a multiple ofrow i m to row j m to clear position ( j m i m − ) , so that the corresponding column operationdoes not introduce another unwanted non-zero entry (as column i m is zero). Again, theresult is a new matrix identical to the original A but with one fewer non-zero element incolumn 1. This completes Step 1. Details for Step 2:
From Step 1, the first column of A has only one non-zero entry a j .So scale the first column/row (and then correct the damage to position (1 n ) by scalingthe n -th column/row to assume a j = 1 . This does not change any other entry of A . Wecan therefore assume that A = | e j B | where B is in sparse form and rank r − , and the first column of A is the unit column e j with j ≥ r .We now show that if j > r , then a carefully constructed sequence of variable swaps willmove the non-zero entry in column 1 from row j to row j − , while keeping the rest ofthe matrix sparse. By induction, this will complete Step 2.To construct the sequence of operations, fix the two sequences j − i , . . . , i m and j = j , . . . , j m satisfying the conditions of Lemma 7.6. Let m = min { m , m } . We claim that thesequence of m swaps x i ℓ ↔ x j ℓ (while fixing all other variables) for ℓ = 1 , , . . . , m willhave the desired effect of replacing e j by e j − as the first column of matrix A , and replacing B by some sparse B ′ of the same rank.The key observation is that if B is in sparse form, then swapping a zero column withan adjacent column will produce a new matrix B ′ also in sparse form. Moreover, a matrixin sparse form can have non-zero adjacent columns with non-zero entries only in adjacent rows. Both these facts follow easily from property (3) of sparse form.Now to check the claim, observe that swapping the variables indexed j and i movesthe non-zero entry of column 1 from row j to (the adjacent) row j − as desired. Butit also swaps the adjacent columns j and i . If either of these is a zero column, thenew "middle matrix" B ′ is in sparse form and the proof is complete. Otherwise, we havealso swapped column vectors e i and e j (which occupy the adjacent columns indexed j and i in A ), destroying the "anti-diagonal" property (3) of the sparse matrix B . In thiscase, however, we move on to swap the next set of variables (indexed j and i ). The rowoperation (swapping adjacent rows i and j ) corrects the order of the two standard unitcolumns e i and e j , but—unless one of the two columns i or j is zero—the correspondingcolumn operation again upsets the order of the standard unit column vectors e i and e j which occupy those columns. We repeat this procedure—swapping the variables indexed ℓ and j ℓ — until eventually (at least) one of column i m or j m is zero, and the processterminates with a "middle matrix" B ′ in sparse form, and first column of A ′ the standardcolumn e j − . This completes Step 2, by induction.To summarize, Steps 1 and 2 together show that by changing coordinates, we can put A into the form: e r B where B is in sparse form. This means that A is also sparse—that is, A satisfies theconclusion of Theorem 7.2. The proof of Theorem 7.2 is complete, once we have provedLemma 7.6. (cid:3) Proof of Lemma 7.6.
Fix i ∈ { r, . . . , n − } . The needed sequence is constructed asfollows:(i) If column i is , then the sequence terminates immediately: the one-elementsequence { i } satisfies the needed conditions (1) through (4).(ii) If column i is not zero, then because B is sparse, column i of A has a unique non-zero entry. Let i be its row index, and note that i ∈ { , . . . , r − } . This saysmatrix A is non-zero in position ( i i ) , and conditions (1), (2) and (3) of Lemma7.6 are satisfied for the sequence { i , i } .Now, assume inductively, that we have a sequence i , . . . , i ℓ satisfying (1), (2) and (3) of the lemma, with ℓ ≥ . Now again, if column i ℓ is , we aredone: the complete sequence is { i , . . . , i ℓ } . Otherwise, column i ℓ is not zero, so has aunique non-zero entry in some row indexed i ℓ +1 . Note that i ℓ +1 ∈ { , . . . , r − } , because i ℓ +1 indexes a non-zero row of the matrix B , which is in sparse form (7.2). So i ℓ +1 = i .But also i ℓ +1 must be distinct from all the prior indices i k in the sequence: if i ℓ +1 = i k forsome k with ≤ k < ℓ , condition (1) implies the entries of matrix A in positions ( i k , i k − ) and ( i k , i ℓ ) would both be non-zero, contrary to the sparseness of B unless i k − = i ℓ . Soby induction, we see that the sequence i , . . . , i ℓ , i ℓ +1 satisfies conditions (1), (2) and (3) of the lemma, as well. Clearly, iterating this process,we must terminate in a zero column as we eventually exhaust all the row indices (indeed,after at most r steps).Finally, note that if i = j are two row indices in the range { r, . . . , n − } , the sets { i , i , . . . , i m } and { j , j , . . . , j m ′ } are disjoint by a similar argument: pick the smallestindex ℓ such that i ℓ = j k for some k , and note this implies i ℓ − = j k − . (cid:3) Remark 7.7.
Theorem 7.1 can be compared to an analogous theorem of Bose andChakravarti for Hermitian forms (see definition in Remark 5.6), which says that theHermitian forms are uniquely determined by their rank, which is always equal to their mbedding dimension: up to changing coordinates, we have only x q +11 + · · · + x q +1 r . Inparticular, there are many more Frobenius forms than Hermitian forms.8. Geometric Properties of Extremal Singularities
Smooth projective varieties defined by (certain special) Frobenius forms have long beenunderstood to be extremal in various ways, going at least back to Beniamino Segre [Seg65].It is easy to see that they contain many linear subspaces, for example, which can be usedto show that they are extremal from the point of view of containing rational points; see[Kol15], [BC66] and [HK16].In this section, we collect a few interesting properties of extremal hypersurfaces. By extremal hypersurface , we mean a projective hypersurface defined by a (not necessarilyreduced) Frobenius form; in the reduced case, it is equivalent to ask that the affine coneover it is an extremal singularity.8.1.
Hyperplane Sections.
It is easy to see that every hyperplane section of a extremalhypersurface is extremal. Somewhat surprisingly, the converse is also true:
Theorem 8.2. If X is an extremal hypersurface, then so is every hyperplane section.Conversely, if the ground field is algebraically closed and n ≥ , then any hypersurface X ⊂ P n with the property that all its hyperplane sections are extremal must itself beextremal.Proof. Since both degree and inclusion in m [ p e ] are preserved under taking the quotientby a linear form, the first statement is clear.For the converse, we set up some notation. For a form f and a linear form L , let f denote the form f mod L in the polynomial ring k [ x , x , . . . , x n ] / h L i . Now suppose that f is the defining equation of the hypersurface X with the propertythat every hyperplane section is extremal. Then f is a Frobenius form (for all choices of L ) so f has degree p e + 1 for some e .Fixing coordinates { x , x , . . . , x n } , we can write f uniquely as f = P ni =0 x p e i L i + g ,where the L i are linear forms, and g is some form none of whose monomials are divisibleby x p e i for any i . We need to show that g is zero. For this, it suffices to show that g isdivisible by infinitely many distinct (up to scalar multiple) linear forms.Fix any linear form L . By hypothesis, f is a Frobenius form. Since the set of Frobeniusforms is closed under addition, also g is a Frobenius form. Now if L = x i , the restrictionon the monomials of g implies that g = 0 —that is, that x i divides g for each i . So withoutloss of generality g = ( x x . . . x n ) h, where h is a form of degree p e + 1 − ( n + 1) . ext, we consider what happens when L = x − cx for some c ∈ k . Using theisomorphism k [ x , x , . . . , x n ] / h x − cx i −→ k [ y , . . . , y n ] ( x cy x i y i i ≥ we see that because g mod L is a Frobenius form, also y y . . . y n ˜ h ∈ h y p e , . . . , y p e n i where ˜ h denotes the image of h in the polynomial ring k [ y , . . . , y n ] . Because y , . . . , y n form a regular sequence, this yields (see Lemma 3.5) ˜ h ∈ h y p e − , y p e − , . . . , y p e − n i . But the degree of ˜ h is p e − n which is strictly less than p e − . So h = 0 . In other words, x − cx divides h . Since c was an arbitrary element of k , h must have at least | k | distinctlinear factors. Since k is infinite, the proof is complete. (cid:3) Corollary 8.3. If X is a smooth extremal hypersurface over an algebraically closed field,then all smooth hyperplane sections are isomorphic.Proof. The hyperplane sections of X are extremal by Theorem 8.2. So the smooth hyper-plane sections are cones over full rank extremal singularities, and hence all projectivelyequivalent to the diagonal hypersurface P ni =1 x p e +1 i by Theorem 6.1. (cid:3) The converse of Corollary 8.3 is a theorem of Beauville [Bea90]; restated in our language,it says that if a smooth projective hypersurface X has the property that the smoothhyperplane sections are isomorphic to each other, then X is an extremal hypersurface.8.4. Gauss Map.
Fix an algebraically closed field. Consider a reduced closed subscheme X ⊂ P n of equi-dimension d . The Gauss map of X is the rational map X G ( d, P n ) x T x X sending each smooth point x to its embedded projective tangent space T x X , considered asa point in the Grassmannian of d -dimensional linear subspaces of P n . For a hypersurface X = V ( f ) ⊂ P n defined by a reduced form f , the Gauss map can be described explicitlyas X ( P n ) ∗ x (cid:20) ∂f∂x : ∂f∂x : · · · : ∂f∂x n (cid:21) . This is undetermined along the singular locus of X .It is not hard to see that the Gauss map is finite when X is smooth (without linearcomponents, which would contract to points under the Gauss map). More generally, the(closure of the) image of the Gauss map has dimension dim X − dim Sing( X ) − [Zak93,2.8].In characteristic zero, the Gauss map of a smooth projective variety is birational, butthis can fail in characteristic p . Many authors have studied the question of precisely how this failure happens, eventually realizing that (at least for hypersurfaces), the issue ppears to be only the inseparability of the Gauss map; see [Wal56], [KP91], or [Kaj89]for example.A smooth extremal hypersurface has the property that its Gauss map is highly inseparable—purely inseparable of maximal degree—and its dual hypersurface is also extremal. Thefollowing straightforward statement may be folklore among experts, but we have not foundit simply stated in the literature: Proposition 8.5.
A smooth extremal hypersurface of degree p e + 1 and dimension d hasa purely inseparable Gauss map of degree q d . The dual hypersurface (that is, the imageunder the Gauss map) is also a smooth extremal hypersurface of the same degree. By purely inseparable , here, we mean that the induced map on generic stalks is a purelyinseparable field extension. Proof.
Suppose X = V ( h ) is an extremal hypersurface in P n . Write h = x p e L + x p e L + · · · + x p e n L n = ( ~x [ p e ] ) tr A ~x.
The Gauss map is x (cid:20) ∂h∂x : ∂h∂x : · · · : ∂h∂x n (cid:21) = " n X i =0 a i x p e i : n X i =0 a i x p e i : · · · : n X i =0 a in x p e i = [ x p e : x p e : · · · : x p e n ] A, where A is the (invertible) matrix representing the Frobenius form h . So the Gauss mapfactors as [ x : x : · · · : x n ] [ x p e : x p e : · · · : x p e n ] [ x p e : x p e : · · · : x p e n ] A. Since A is just a linear change of coordinates, we can analyze the induced map on thegeneric stalk for the map [ x : x : · · · : x n ] [ x p e : x p e : · · · : x p e n ] only. Withoutloss of generality, the generic stalk is the fraction field of k h x x , . . . x n x i / D hx pe +10 E , whichis a purely transcendental extension of k of transcendence degree n − generated by therational functions x x , . . . , x n − x . So the Gauss map on stalks can be viewed as simply theinclusion k (cid:16) ( x x ) p e , . . . , ( x n − x ) p e (cid:17) ⊂ k ( x x , . . . , x n − x ) , which is purely inseparable of degree ( p e ) n − where n − is the dimension of the hypersurface.To see that the image is extremal, note because the matrix A is invertible, it sufficesto show the p e -th power map on the homogeneous coordinates has extremal image. Butthe relation h = ( ~x [ p e ] ) tr A ~x on the homogeneous coordinates of X implies the relation (( ~x [ p e ] ) [ p e ] ) tr A [ p e ] ~x [ p e ] on the coordinates [ x p e : x p e : · · · : x p e n ] of the image. So the imageis isomorphic to the extremal singularity defined by the Frobenius form represented by A [ p e ] . (cid:3) Remark 8.6.
Conjecture 2 in [KP91] can be interpreted as predicting that any smoothhypersurface of degree d ≥ with the property that its dual hypersurface is smooth must e defined by a Frobenius form. (This is known for curves [Hom87, 6.1, 6.7], [Hef89, 7.8]and surfaces [KP91, 14].) Thus, in light of Proposition 8.5, we should expect a smoothhypersurface is extremal if and only if it dual hypersurface is smooth. Remark 8.7.
Even if the extremal hypersurface X is not smooth, the proof of Proposition8.5 shows that the Gauss map of an extremal hypersurface is the p e -th power map followedby a linear projection.8.8. Lines on extremal hypersurfaces.
Extremal hypersurfaces are extremal also inthe behavior of the linear subspaces they contain; see, for example, the discussion in[Kol15, §35]. One simple way to describe this is by looking at the special configurationsof intersecting lines on them.
Definition 8.9.
A configuration of lines in the projective plane is perfect star of degree d ≥ if it projectively equivalent to d reduced concurrent lines with slopes rangingthrough the d -th roots of unity. Equivalently, a perfect star of degree d is defined by anequation x d − y d , where the characteristic of the ground field does not divide d . Remark 8.10.
We could include d = 1 , in Definition 8.9, but then every configurationof lines d forms a perfect star. When d = 3 , a configuration of lines is a perfect star ifand only if the three lines are concurrent. The condition becomes more restrictive as d gets larger.Perfect stars are clearly very special configurations of lines—we don’t expect mosthypersurfaces to contain any , unless the hypersurface contains an entire plane. So thefollowing result emphasizes that extremal hypersurfaces really have extremal behavior interms of the configuration of lines they contain: Proposition 8.11.
Let X ⊂ P n be an extremal hypersurface of degree q + 1 where q isa power of the characteristic p > . Suppose ℓ and ℓ are intersecting lines contained in X , and let Λ be the plane they span. If Λ is not contained in X , then the plane section Λ ∩ X is either a perfect star of degree q + 1 or the union of a q -fold line and a reducedline.Proof. If Λ does not lie on X , then Λ ∩ X ⊂ Λ ∼ = P is a extremal curve by Theorem8.2. Choose coordinates for P so that ℓ and ℓ are given by the vanishing of x and y in Λ ∼ = P . The form h defining the plane section with Λ is h = x q L + y q L + z q L Because ℓ and ℓ lie on this curve, we know both x and y divide h . This forces y | L , x | L , and xy | L . As the L i are linear forms, the form h is h = ax q y + by q x = xy ( ax q − + by q − ) for some scalars a, b . So h factors into q + 1 linear forms, all distinct unless one of a or b is zero. In the former case, we can scale x and y to assume a = 1 and b = − to get aperfect star and in the latter case, we have the non-reduced line configuration defined by x q y . (cid:3) eferences [AH19] J. D. Achter and E. W. Howe, Hasse–witt and cartier–manin matrices: a warning and arequest , Contemp. Math. (2019), 1–18.[BC66] R. C. Bose and I. M. Chakravarti,
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E-mail address : [email protected]@umich.edu